{"id":2909,"date":"2026-01-19T15:58:42","date_gmt":"2026-01-19T10:28:42","guid":{"rendered":"https:\/\/www.manabadi.co.in\/boards\/?p=2909"},"modified":"2026-01-19T15:58:44","modified_gmt":"2026-01-19T10:28:44","slug":"ts-inter-1st-year-maths-1a-addition-of-vectors-solutions-exercise-4b","status":"publish","type":"post","link":"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-addition-of-vectors-solutions-exercise-4b\/","title":{"rendered":"TS Inter 1st Year Maths 1A Addition of Vectors Solutions Exercise 4(b)"},"content":{"rendered":"\n<h3>I.<br> Question 1.<br> Find the vector equation of the line passing through the point 2i\u0305 + 3j\u0305 + k\u0305 and parallel to the vector 4i\u0305 \u2013 2j\u0305 + 3k\u0305.(March 2015-A.P) (May, March \u201901) (V.S.A)<\/h3> <p>Answer:<br> Let a = 2i\u0305 + 3j\u0305 + k\u0305 and b = 4i\u0305 \u2013 2 j\u0305 + 3k\u0305<br> The vector equation of the line passing through the point a\u0305 and parallel to the vector b\u0305 is<br> r\u0305 = a\u0305 + tb\u0305 where t is a scalar.<br> r\u0305 = (2i\u0305 + 3j\u0305 + k\u0305) + t (4i\u0305 \u2013 2j\u0305 + 3k\u0305)<br> \u21d2 r\u0305 = (2 + 4t) i\u0305 + (3 \u2013 2t) j\u0305 + (1 + 3t) k\u0305<\/p>\n\n<h3>Question 2.<br> OABC is a parallelogram. If <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-1-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-1\" style=\"width: 3.963em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 3.404em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.121em, 1003.4em, 2.472em, -999.998em); top: -2.328em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-2\"><span class=\"munderover\" id=\"MathJax-Span-3\"><span style=\"display: inline-block; position: relative; width: 1.54em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1001.49em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-4\"><span class=\"mrow\" id=\"MathJax-Span-5\"><span class=\"mi\" id=\"MathJax-Span-6\" style=\"font-family: MathJax_Main;\">O<\/span><span class=\"mi\" id=\"MathJax-Span-7\" style=\"font-family: MathJax_Main;\">A<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1001.49em, 3.777em, -999.998em); top: -4.518em; left: 0.002em;\"><span class=\"mo\" id=\"MathJax-Span-8\" style=\"\"><span style=\"display: inline-block; position: relative; width: 1.493em; height: 0px;\"><span style=\"position: absolute; top: -4.005em; left: -0.044em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 1.214em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.096em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.329em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.515em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.701em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.888em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 1.074em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-9\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">=<\/span><span class=\"munderover\" id=\"MathJax-Span-10\" style=\"padding-left: 0.282em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.51em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-11\"><span class=\"mrow\" id=\"MathJax-Span-12\"><span class=\"mi\" id=\"MathJax-Span-13\" style=\"font-family: MathJax_Main;\">a<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1000.47em, 3.777em, -999.998em); top: -4.285em; left: 0.002em;\"><span class=\"mo\" id=\"MathJax-Span-14\" style=\"\"><span style=\"display: inline-block; position: relative; width: 0.468em; height: 0px;\"><span style=\"position: absolute; top: -4.005em; left: -0.044em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.189em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.096em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.332em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.051em; border-left: 0px solid; width: 0px; height: 1.354em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-1\">\\overline{\\mathrm{OA}}=\\overline{\\mathrm{a}}<\/script> and <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-2-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-15\" style=\"width: 3.87em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 3.311em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.167em, 1003.31em, 2.472em, -999.998em); top: -2.328em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-16\"><span class=\"munderover\" id=\"MathJax-Span-17\"><span style=\"display: inline-block; position: relative; width: 1.493em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1001.45em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-18\"><span class=\"mrow\" id=\"MathJax-Span-19\"><span class=\"mi\" id=\"MathJax-Span-20\" style=\"font-family: MathJax_Main;\">O<\/span><span class=\"mi\" id=\"MathJax-Span-21\" style=\"font-family: MathJax_Main;\">C<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1001.49em, 3.777em, -999.998em); top: -4.518em; left: 0.002em;\"><span class=\"mo\" id=\"MathJax-Span-22\" style=\"\"><span style=\"display: inline-block; position: relative; width: 1.493em; height: 0px;\"><span style=\"position: absolute; top: -4.005em; left: -0.044em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 1.167em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.096em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.282em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.468em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.655em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.841em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 1.027em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-23\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">=<\/span><span class=\"munderover\" id=\"MathJax-Span-24\" style=\"padding-left: 0.282em;\"><span style=\"display: inline-block; position: relative; width: 0.468em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.42em, 4.15em, -999.998em); top: -4.005em; left: 0.002em;\"><span class=\"texatom\" id=\"MathJax-Span-25\"><span class=\"mrow\" id=\"MathJax-Span-26\"><span class=\"mi\" id=\"MathJax-Span-27\" style=\"font-family: MathJax_Main;\">c<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1000.47em, 3.777em, -999.998em); top: -4.285em; left: 0em;\"><span class=\"mo\" id=\"MathJax-Span-28\" style=\"\"><span style=\"display: inline-block; position: relative; width: 0.468em; height: 0px;\"><span style=\"position: absolute; top: -4.005em; left: -0.044em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.189em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.332em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.051em; border-left: 0px solid; width: 0px; height: 1.354em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-2\">\\overline{\\mathrm{OC}}=\\overline{\\mathrm{c}}<\/script>. Find the vector equation of the side BC. (March 2015-T.S) (V.S.A)<\/h3>\n\n<p>Answer:<br> OABC is a parallelogram.<br>\n\n<img fetchpriority=\"high\" decoding=\"async\" class=\"alignnone size-full wp-image-6812\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-4-Addition-of-Vectors-Ex-4b-1.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 4 Addition of Vectors Ex 4(b) 1\" width=\"297\" height=\"229\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 4 Addition of Vectors Ex 4(b) 1\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 4 Addition of Vectors Ex 4(a)\"><br> \u2234 The vector equation of side BC is r\u0305 = (1 \u2013 t)c\u0305 + t(a\u0305 + c\u0305)<br> = (1 \u2013 t + t)c\u0305 + ta\u0305<br> = c\u0305 + t a\u0305 where t \u2208 R.<\/p>\n\n<h3>Question 3.<br> If a\u0305, b\u0305, c\u0305 are the position vectors of the vertices A, B and C respectively of a \u0394ABC, then find the vector equation of the median through the vertex A. (March 2013) (V.S.A)<\/h3>\n\n<p>Answer:<br>\n\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-6811\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-4-Addition-of-Vectors-Ex-4b-2.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 4 Addition of Vectors Ex 4(b) 2\" width=\"249\" height=\"216\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 4 Addition of Vectors Ex 4(b) 2\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 4 Addition of Vectors Ex 4(a)\"><br>\n\nLet <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-3-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-29\" style=\"width: 8.437em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 7.272em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.121em, 1007.27em, 2.658em, -999.998em); top: -2.328em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-30\"><span class=\"munderover\" id=\"MathJax-Span-31\"><span style=\"display: inline-block; position: relative; width: 1.54em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1001.49em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-32\"><span class=\"mrow\" id=\"MathJax-Span-33\"><span class=\"mi\" id=\"MathJax-Span-34\" style=\"font-family: MathJax_Main;\">O<\/span><span class=\"mi\" id=\"MathJax-Span-35\" style=\"font-family: MathJax_Main;\">A<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1001.49em, 3.777em, -999.998em); top: -4.518em; left: 0.002em;\"><span class=\"mo\" id=\"MathJax-Span-36\" style=\"\"><span style=\"display: inline-block; position: relative; width: 1.493em; height: 0px;\"><span style=\"position: absolute; top: -4.005em; left: -0.044em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 1.214em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.096em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.329em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.515em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.701em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.888em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 1.074em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-37\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">=<\/span><span class=\"munderover\" id=\"MathJax-Span-38\" style=\"padding-left: 0.282em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.51em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-39\"><span class=\"mrow\" id=\"MathJax-Span-40\"><span class=\"mi\" id=\"MathJax-Span-41\" style=\"font-family: MathJax_Main;\">a<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1000.47em, 3.777em, -999.998em); top: -4.285em; left: 0.002em;\"><span class=\"mo\" id=\"MathJax-Span-42\" style=\"\"><span style=\"display: inline-block; position: relative; width: 0.468em; height: 0px;\"><span style=\"position: absolute; top: -4.005em; left: -0.044em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.189em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.096em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-43\" style=\"font-family: MathJax_Main;\">,<\/span><span class=\"munderover\" id=\"MathJax-Span-44\" style=\"padding-left: 0.189em;\"><span style=\"display: inline-block; position: relative; width: 1.493em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1001.45em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-45\"><span class=\"mrow\" id=\"MathJax-Span-46\"><span class=\"mi\" id=\"MathJax-Span-47\" style=\"font-family: MathJax_Main;\">O<\/span><span class=\"mi\" id=\"MathJax-Span-48\" style=\"font-family: MathJax_Main;\">B<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1001.45em, 3.777em, -999.998em); top: -4.518em; left: 0.002em;\"><span class=\"mo\" id=\"MathJax-Span-49\" style=\"\"><span style=\"display: inline-block; position: relative; width: 1.447em; height: 0px;\"><span style=\"position: absolute; top: -4.005em; left: -0.044em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 1.167em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.142em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.375em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.562em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.795em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.981em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-50\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">=<\/span><span class=\"munderover\" id=\"MathJax-Span-51\" style=\"padding-left: 0.282em;\"><span style=\"display: inline-block; position: relative; width: 0.562em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.51em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-52\"><span class=\"mrow\" id=\"MathJax-Span-53\"><span class=\"mi\" id=\"MathJax-Span-54\" style=\"font-family: MathJax_Main;\">b<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1000.51em, 3.777em, -999.998em); top: -4.518em; left: 0.002em;\"><span class=\"mo\" id=\"MathJax-Span-55\" style=\"\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; top: -4.005em; left: -0.044em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.235em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.096em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.332em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.268em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-3\">\\overline{\\mathrm{OA}}=\\overline{\\mathrm{a}}, \\overline{\\mathrm{OB}}=\\overline{\\mathrm{b}}<\/script>, and <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-4-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-56\" style=\"width: 3.87em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 3.311em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.167em, 1003.31em, 2.472em, -999.998em); top: -2.328em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-57\"><span class=\"munderover\" id=\"MathJax-Span-58\"><span style=\"display: inline-block; position: relative; width: 1.493em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1001.45em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-59\"><span class=\"mrow\" id=\"MathJax-Span-60\"><span class=\"mi\" id=\"MathJax-Span-61\" style=\"font-family: MathJax_Main;\">O<\/span><span class=\"mi\" id=\"MathJax-Span-62\" style=\"font-family: MathJax_Main;\">C<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1001.49em, 3.777em, -999.998em); top: -4.518em; left: 0.002em;\"><span class=\"mo\" id=\"MathJax-Span-63\" style=\"\"><span style=\"display: inline-block; position: relative; width: 1.493em; height: 0px;\"><span style=\"position: absolute; top: -4.005em; left: -0.044em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 1.167em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.096em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.282em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.468em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.655em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.841em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 1.027em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-64\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">=<\/span><span class=\"munderover\" id=\"MathJax-Span-65\" style=\"padding-left: 0.282em;\"><span style=\"display: inline-block; position: relative; width: 0.468em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.42em, 4.15em, -999.998em); top: -4.005em; left: 0.002em;\"><span class=\"texatom\" id=\"MathJax-Span-66\"><span class=\"mrow\" id=\"MathJax-Span-67\"><span class=\"mi\" id=\"MathJax-Span-68\" style=\"font-family: MathJax_Main;\">c<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1000.47em, 3.777em, -999.998em); top: -4.285em; left: 0em;\"><span class=\"mo\" id=\"MathJax-Span-69\" style=\"\"><span style=\"display: inline-block; position: relative; width: 0.468em; height: 0px;\"><span style=\"position: absolute; top: -4.005em; left: -0.044em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.189em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.332em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.051em; border-left: 0px solid; width: 0px; height: 1.354em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-4\">\\overline{\\mathrm{OC}}=\\overline{\\mathrm{c}}<\/script><br> Vector equation of the median AD is (1 \u2013 t)<br> a\u0305 + tb\u0305 = r\u0305<br> r\u0305 = (1 \u2013 t)a\u0305 + t<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-5-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-70\" style=\"width: 3.357em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 2.891em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.866em, 1002.71em, 3.963em, -999.998em); top: -3.166em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-71\"><span class=\"mrow\" id=\"MathJax-Span-72\"><span class=\"mo\" id=\"MathJax-Span-73\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size2;\">(<\/span><\/span><span class=\"mfrac\" id=\"MathJax-Span-74\"><span style=\"display: inline-block; position: relative; width: 1.4em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.124em, 1001.26em, 4.196em, -999.998em); top: -4.471em; left: 50%; margin-left: -0.603em;\"><span class=\"mrow\" id=\"MathJax-Span-75\"><span class=\"munderover\" id=\"MathJax-Span-76\"><span style=\"display: inline-block; position: relative; width: 0.375em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.357em, 1000.38em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-77\"><span class=\"mrow\" id=\"MathJax-Span-78\"><span class=\"mi\" id=\"MathJax-Span-79\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">b<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.497em, 1000.38em, 3.87em, -999.998em); top: -4.378em; left: 0.002em;\"><span class=\"mo\" id=\"MathJax-Span-80\" style=\"\"><span style=\"display: inline-block; position: relative; width: 0.375em; height: 0px;\"><span style=\"position: absolute; top: -4.005em; left: -0.044em;\"><span style=\"font-size: 50%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.189em;\"><span style=\"font-size: 50%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.096em;\"><span style=\"font-size: 50%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-81\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">+<\/span><span class=\"munderover\" id=\"MathJax-Span-82\"><span style=\"display: inline-block; position: relative; width: 0.329em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.544em, 1000.28em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-83\"><span class=\"mrow\" id=\"MathJax-Span-84\"><span class=\"mi\" id=\"MathJax-Span-85\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">c<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.497em, 1000.33em, 3.87em, -999.998em); top: -4.192em; left: 0.002em;\"><span class=\"mo\" id=\"MathJax-Span-86\" style=\"\"><span style=\"display: inline-block; position: relative; width: 0.329em; height: 0px;\"><span style=\"position: absolute; top: -4.005em; left: -0.044em;\"><span style=\"font-size: 50%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.096em;\"><span style=\"font-size: 50%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.049em;\"><span style=\"font-size: 50%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-87\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1001.4em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.4em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-88\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size2;\">)<\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 3.171em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.808em; border-left: 0px solid; width: 0px; height: 2.219em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-5\">\\left(\\frac{\\overline{\\mathrm{b}}+\\overline{\\mathrm{c}}}{2}\\right)<\/script><\/p>\n\n<h3>Question 4.<br> Find the vector equation of the line joining the points 2i\u0305 + j\u0305 + 3k\u0305 and \u2013 4i\u0305 + 3j\u0305 \u2013 k\u0305.(V.S.A)<\/h3>\n\n<p>Answer:<br> Let a = 2i\u0305 + j\u0305 + 3k\u0305 and b = -4i\u0305 + 3 j\u0305 \u2013 k\u0305<br> The vector equation of the line passing through the points a,b is<br> r\u0305 = (1 \u2013 t)a\u0305 + tb\u0305, t \u2208 R<br> = a\u0305 + t (b\u0305 \u2013 a\u0305)<br> = (2i\u0305 + j\u0305 + 3k\u0305) + t (-4i\u0305 + 3j\u0305 \u2013 k\u0305 \u2013 2i\u0305 \u2013 j\u0305 \u2013 3k\u0305)<br> = (2i\u0305 + j\u0305 + 3k\u0305) +t(-6i\u0305 + 2j\u0305 \u2013 4k\u0305)<\/p>\n\n<h3>Question 5.<br> Find the vector equation of the plane passing through the points i\u0305 \u2013 2j\u0305 + 5k\u0305, \u2013 5j\u0305 \u2013 k\u0305 and -3 i\u0305 + 5j\u0305. (V.S.A)<\/h3>\n\n<p>Answer:<br> Let a\u0305 = i\u0305 \u2013 2 j\u0305 + 5k\u0305, b\u0305 = -5 j\u0305 \u2013 k\u0305, c = -3i\u0305 + 5j\u0305. (May 2014)<br> The vector equation of the plane passing through the points a\u0305, b\u0305, c\u0305 is r = (1 \u2013 s \u2013 t)a\u0305 + sb\u0305 + tc\u0305 where s, t \u2208 R<br> = a\u0305 + s(b\u0305 \u2013 a\u0305) + t(c\u0305 \u2013 a\u0305)<br> = (i\u0305 \u2013 2j\u0305 + 5k\u0305) + s(-5j\u0305 \u2013 k\u0305 \u2013 i\u0305 + 2j\u0305 \u2013 5k\u0305) + t(-3i\u0305 + 5j\u0305 \u2013 i\u0305 + 2j\u0305 \u2013 5k\u0305)<br> = i\u0305 \u2013 2j\u0305 + 5k\u0305 + s(-i\u0305 \u2013 3j\u0305 \u2013 6k\u0305) + t(-4i\u0305 + 7j\u0305 \u2013 5k\u0305)<\/p>\n\n<h3>Question 6.<br> Find the vector equation of the plane passing through the points (0,0, 0), (0, 5, 0) and (2, 0, 1). (V.S.A)<\/h3>\n\n<p>Answer:<br> The vector equation of the plane through a, b,c is<br> r\u0305 = (1 \u2013 s \u2013 t)a\u0305 + sb\u0305 + tc\u0305 where s, t \u2208 R<br> \u21d2 r\u0305 = (1 \u2013 s \u2013 t) 0 + s(5j\u0305) + t(2i\u0305 + k\u0305)<br> = (5s) j\u0305 + t(2i\u0305 + k\u0305);s, t \u2208 R<\/p>\n\n<h3>II.<br> Question 1.<br> If a, b, c are noncoplanar find the point of intersection of the line passing through the points 2a\u0305 + 3b\u0305 \u2013 c\u0305, 3a\u0305 + 4b\u0305 \u2013 2c\u0305 with the line joining points a\u0305 \u2013 2b\u0305 + 3c\u0305, a\u0305 \u2013 6b\u0305 + 6c\u0305. (S.A)<\/h3>\n\n<p>Answer:<br> The vector equation of the straight line passing through the points 2a\u0305 + 3b\u0305 \u2013 c\u0305 and 3a\u0305 + 4b\u0305 \u2013 2c\u0305 is<br> r\u0305 = (1 \u2013 t) (2a\u0305 + 3b\u0305 \u2013 c\u0305) + t(3a\u0305 + 4b\u0305 \u2013 2c\u0305) where t \u2208 1<br> \u21d2 r\u0305 = (2 + t)a\u0305 + (3 + t) b\u0305 + (-1 \u2013 t)c\u0305<br> = (2a\u0305 + 3b\u0305 \u2013 c\u0305) + t (a\u0305 + b\u0305 \u2013 c\u0305) \u2026\u2026\u2026\u2026\u2026(1)<br> The vector equation of the straight line passing through the points a\u0305 \u2013 2b\u0305 + 3c\u0305 and a\u0305 \u2013 6b\u0305 + 6c\u0305 is<br> r\u0305 = (a\u0305 \u2013 2b\u0305 + 3c\u0305) (1 \u2013 s) + s (a\u0305 \u2013 6b\u0305 + 6c\u0305) where s \u2208 R<br> \u21d2 r\u0305 = a\u0305 + (-2 \u2013 4s) b\u0305 + (3 + 3s)c\u0305<br> = (a\u0305 \u2013 2b\u0305 + 3c\u0305) + s (-4b\u0305 + 3c\u0305) \u2026\u2026\u2026\u2026..(2)<br> Equating coefficients a\u0305, b\u0305, c\u0305 in (1) and (2) we have<br> 2 + t = 1 \u2026\u2026\u2026..(3)<br> 3 + t = \u2013 2 \u2013 4s \u2026\u2026\u2026.(4)<br> and \u2013 1 \u2013 t = 3 + 3s \u2026\u2026\u2026..(5)<br> Solving equations (3), (4) and (5) we get t = \u2013 1, and s = \u2013 1<br> Hence from (1) and (2) the point of intersection of lines (1) and (2) is a\u0305 + 2b\u0305<br> Also line (1) is parallel to a + b \u2013 c ancl (2) is parallel to -4b\u0305 + 3c\u0305<br> If a\u0305 + b\u0305 \u2013 c\u0305 and 3c\u0305 \u2013 4b\u0305 are parallel then two lines are same since they have common point otherwise they have only one point of intersection a\u0305 + 2b\u0305<\/p>\n\n<h3>Question 2.<br> ABCD is a trapezium in which AB and CD are parallel. Prove by vector methods that the mid points of the sides AB, CD and the intersection of the diagonals are collinear. (E.Q)<\/h3>\n\n<p>Answer:<br> Let A be the origin and <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-6-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-89\" style=\"width: 1.68em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.447em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.074em, 1001.45em, 2.425em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-90\"><span class=\"munderover\" id=\"MathJax-Span-91\"><span style=\"display: inline-block; position: relative; width: 1.447em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1001.4em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-92\"><span class=\"mrow\" id=\"MathJax-Span-93\"><span class=\"mi\" id=\"MathJax-Span-94\" style=\"font-family: MathJax_Main;\">A<\/span><span class=\"mi\" id=\"MathJax-Span-95\" style=\"font-family: MathJax_Main;\">B<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1001.45em, 3.777em, -999.998em); top: -4.518em; left: 0.002em;\"><span class=\"mo\" id=\"MathJax-Span-96\" style=\"\"><span style=\"display: inline-block; position: relative; width: 1.447em; height: 0px;\"><span style=\"position: absolute; top: -4.005em; left: -0.044em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 1.121em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.142em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.329em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.562em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.748em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.981em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.051em; border-left: 0px solid; width: 0px; height: 1.354em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-6\">\\overline{\\mathrm{AB}}<\/script> = b\u0305<br> \u2234 <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-7-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-97\" style=\"width: 1.726em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.493em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.121em, 1001.49em, 2.425em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-98\"><span class=\"munderover\" id=\"MathJax-Span-99\"><span style=\"display: inline-block; position: relative; width: 1.493em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1001.45em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-100\"><span class=\"mrow\" id=\"MathJax-Span-101\"><span class=\"mi\" id=\"MathJax-Span-102\" style=\"font-family: MathJax_Main;\">D<\/span><span class=\"mi\" id=\"MathJax-Span-103\" style=\"font-family: MathJax_Main;\">C<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1001.45em, 3.777em, -999.998em); top: -4.518em; left: 0.002em;\"><span class=\"mo\" id=\"MathJax-Span-104\" style=\"\"><span style=\"display: inline-block; position: relative; width: 1.447em; height: 0px;\"><span style=\"position: absolute; top: -4.005em; left: -0.044em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 1.167em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.142em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.375em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.562em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.795em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.981em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.051em; border-left: 0px solid; width: 0px; height: 1.354em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-7\">\\overline{\\mathrm{DC}}<\/script> = sb\u0305 (\u2235 <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-8-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-105\" style=\"width: 4.01em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 3.451em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.121em, 1003.45em, 2.705em, -999.998em); top: -2.328em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-106\"><span class=\"munderover\" id=\"MathJax-Span-107\"><span style=\"display: inline-block; position: relative; width: 1.493em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1001.45em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-108\"><span class=\"mrow\" id=\"MathJax-Span-109\"><span class=\"mi\" id=\"MathJax-Span-110\" style=\"font-family: MathJax_Main;\">D<\/span><span class=\"mi\" id=\"MathJax-Span-111\" style=\"font-family: MathJax_Main;\">C<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1001.45em, 3.777em, -999.998em); top: -4.518em; left: 0.002em;\"><span class=\"mo\" id=\"MathJax-Span-112\" style=\"\"><span style=\"display: inline-block; position: relative; width: 1.447em; height: 0px;\"><span style=\"position: absolute; top: -4.005em; left: -0.044em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 1.167em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.142em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.375em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.562em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.795em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.981em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-113\" style=\"font-family: MathJax_Main;\">\u2225<\/span><span class=\"munderover\" id=\"MathJax-Span-114\"><span style=\"display: inline-block; position: relative; width: 1.447em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1001.4em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-115\"><span class=\"mrow\" id=\"MathJax-Span-116\"><span class=\"mi\" id=\"MathJax-Span-117\" style=\"font-family: MathJax_Main;\">A<\/span><span class=\"mi\" id=\"MathJax-Span-118\" style=\"font-family: MathJax_Main;\">B<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1001.45em, 3.777em, -999.998em); top: -4.518em; left: 0.002em;\"><span class=\"mo\" id=\"MathJax-Span-119\" style=\"\"><span style=\"display: inline-block; position: relative; width: 1.447em; height: 0px;\"><span style=\"position: absolute; top: -4.005em; left: -0.044em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 1.121em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.142em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.329em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.562em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.748em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.981em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.332em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.322em; border-left: 0px solid; width: 0px; height: 1.624em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-8\">\\overline{\\mathrm{DC}} \\| \\overline{\\mathrm{AB}}<\/script>)<br> <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-9-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-120\" style=\"width: 1.726em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.493em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.121em, 1001.49em, 2.425em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-121\"><span class=\"munderover\" id=\"MathJax-Span-122\"><span style=\"display: inline-block; position: relative; width: 1.493em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1001.45em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-123\"><span class=\"mrow\" id=\"MathJax-Span-124\"><span class=\"mi\" id=\"MathJax-Span-125\" style=\"font-family: MathJax_Main;\">D<\/span><span class=\"mi\" id=\"MathJax-Span-126\" style=\"font-family: MathJax_Main;\">C<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1001.45em, 3.777em, -999.998em); top: -4.518em; left: 0.002em;\"><span class=\"mo\" id=\"MathJax-Span-127\" style=\"\"><span style=\"display: inline-block; position: relative; width: 1.447em; height: 0px;\"><span style=\"position: absolute; top: -4.005em; left: -0.044em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 1.167em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.142em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.375em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.562em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.795em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.981em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.051em; border-left: 0px solid; width: 0px; height: 1.354em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-9\">\\overline{\\mathrm{DC}}<\/script> = c\u0305 \u2013 d\u0305 = sb<br> \u21d2 d\u0305 = c\u0305 \u2013 sb\u0305<br> \u21d2 c\u0305 \u2013 d\u0305 = sb\u0305<br>\n\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-6810\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-4-Addition-of-Vectors-Ex-4b-3.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 4 Addition of Vectors Ex 4(b) 3\" width=\"333\" height=\"384\" sizes=\"auto, (max-width: 333px) 100vw, 333px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 4 Addition of Vectors Ex 4(b) 3\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 4 Addition of Vectors Ex 4(a)\"><br> Equation of diagonal AC is<br> r\u0305 = (1 \u2013 t)o + tc\u0305<br> = tc\u0305 for t \u2208 R \u2026\u2026\u2026\u2026..(2)<br> Equation of diagonal BD is<br> r\u0305 = (1 \u2013 s)b\u0305 + sd\u0305 for s \u2208 R \u2026\u2026\u2026\u2026(3)<\/p>\n\n<p>Let R be the point of intersection of diagonals AC and BD.<br> From (2) and (3) t c = (1 \u2013 s) b\u0305 + s d\u0305<br> \u21d2 t c\u0305 = (1 \u2013 s) b\u0305 + s (c \u2013 sb\u0305) from (1)<br> \u21d2 tc\u0305 = (1 \u2013 s)\u03bb (c \u2013 d\u0305) + sd\u0305<br> = (1 \u2013 s) \u03bbc\u0305 \u2013 [\u03bb(1 \u2013 s) \u2013 s]d\u0305<br> Equating coefficients of c\u0305 and d\u0305 on both sides<br> t = (1 \u2013 s) \u03bb and \u03bb (1 \u2013 s) \u2013 s = 0<br> \u21d2 s = \u03bb (1 \u2013 s)<br> \u21d2 s(1 + \u03bb) = \u03bb<br> \u21d2 s = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-10-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-128\" style=\"width: 2.006em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.726em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.26em, 1001.73em, 2.891em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-129\"><span class=\"mfrac\" id=\"MathJax-Span-130\"><span style=\"display: inline-block; position: relative; width: 1.447em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.357em, 1000.38em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mi\" id=\"MathJax-Span-131\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">\u03bb<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1001.31em, 4.196em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.65em;\"><span class=\"mrow\" id=\"MathJax-Span-132\"><span class=\"mn\" id=\"MathJax-Span-133\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span class=\"mo\" id=\"MathJax-Span-134\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">+<\/span><span class=\"mi\" id=\"MathJax-Span-135\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">\u03bb<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1001.45em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.447em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.592em; border-left: 0px solid; width: 0px; height: 1.678em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-10\">\\frac{\\lambda}{1+\\lambda}<\/script><br> \u2234 t = (1 \u2013 s)\u03bb = (1 \u2013 <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-11-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-136\" style=\"width: 2.006em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.726em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.26em, 1001.73em, 2.891em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-137\"><span class=\"mfrac\" id=\"MathJax-Span-138\"><span style=\"display: inline-block; position: relative; width: 1.447em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.357em, 1000.38em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mi\" id=\"MathJax-Span-139\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">\u03bb<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1001.31em, 4.196em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.65em;\"><span class=\"mrow\" id=\"MathJax-Span-140\"><span class=\"mn\" id=\"MathJax-Span-141\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span class=\"mo\" id=\"MathJax-Span-142\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">+<\/span><span class=\"mi\" id=\"MathJax-Span-143\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">\u03bb<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1001.45em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.447em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.592em; border-left: 0px solid; width: 0px; height: 1.678em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-11\">\\frac{\\lambda}{1+\\lambda}<\/script>)\u03bb<br> = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-12-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-144\" style=\"width: 2.006em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.726em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.26em, 1001.73em, 2.891em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-145\"><span class=\"mfrac\" id=\"MathJax-Span-146\"><span style=\"display: inline-block; position: relative; width: 1.447em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.357em, 1000.38em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mi\" id=\"MathJax-Span-147\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">\u03bb<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1001.31em, 4.196em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.65em;\"><span class=\"mrow\" id=\"MathJax-Span-148\"><span class=\"mn\" id=\"MathJax-Span-149\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span class=\"mo\" id=\"MathJax-Span-150\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">+<\/span><span class=\"mi\" id=\"MathJax-Span-151\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">\u03bb<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1001.45em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.447em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.592em; border-left: 0px solid; width: 0px; height: 1.678em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-12\">\\frac{\\lambda}{1+\\lambda}<\/script><br> Position vector of the point of intersection \u2018R\u2019 is<br>\n\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-6809\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-4-Addition-of-Vectors-Ex-4b-4.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 4 Addition of Vectors Ex 4(b) 4\" width=\"329\" height=\"1033\" sizes=\"auto, (max-width: 329px) 100vw, 329px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 4 Addition of Vectors Ex 4(b) 4\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 4 Addition of Vectors Ex 4(a)\"><br>\n\nFrom (4) and (5)<br> <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-13-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-152\" style=\"width: 1.913em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.633em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.121em, 1001.63em, 2.425em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-153\"><span class=\"munderover\" id=\"MathJax-Span-154\"><span style=\"display: inline-block; position: relative; width: 1.633em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1001.63em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-155\"><span class=\"mrow\" id=\"MathJax-Span-156\"><span class=\"mi\" id=\"MathJax-Span-157\" style=\"font-family: MathJax_Main;\">R<\/span><span class=\"mi\" id=\"MathJax-Span-158\" style=\"font-family: MathJax_Main;\">M<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1001.63em, 3.777em, -999.998em); top: -4.518em; left: 0.002em;\"><span class=\"mo\" id=\"MathJax-Span-159\" style=\"\"><span style=\"display: inline-block; position: relative; width: 1.633em; height: 0px;\"><span style=\"position: absolute; top: -4.005em; left: -0.044em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 1.307em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.142em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.329em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.562em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.748em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.981em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 1.167em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.051em; border-left: 0px solid; width: 0px; height: 1.3em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-13\">\\overline{\\mathrm{RM}}<\/script> = \u03bb <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-14-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-160\" style=\"width: 1.726em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.493em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.121em, 1001.49em, 2.425em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-161\"><span class=\"munderover\" id=\"MathJax-Span-162\"><span style=\"display: inline-block; position: relative; width: 1.493em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1001.49em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-163\"><span class=\"mrow\" id=\"MathJax-Span-164\"><span class=\"mi\" id=\"MathJax-Span-165\" style=\"font-family: MathJax_Main;\">N<\/span><span class=\"mi\" id=\"MathJax-Span-166\" style=\"font-family: MathJax_Main;\">R<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1001.45em, 3.777em, -999.998em); top: -4.518em; left: 0.002em;\"><span class=\"mo\" id=\"MathJax-Span-167\" style=\"\"><span style=\"display: inline-block; position: relative; width: 1.447em; height: 0px;\"><span style=\"position: absolute; top: -4.005em; left: -0.044em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 1.167em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.142em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.375em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.562em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.795em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.981em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.051em; border-left: 0px solid; width: 0px; height: 1.3em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-14\">\\overline{\\mathrm{NR}}<\/script><br> \u21d2 M, R, N are collinear.<br> So the mid points of parallel sides of a trapezium and the point of intersection of the diagonals are collinear.<\/p>\n\n<h3>Question 3.<br> In a quadrilateral ABCD, if the midpoints of one pair of opposite sides and the point of intersection of the diagonals are collinear, using vector methods, prove that the quadrilateral ABCD is a trapezium. (S.A)<\/h3>\n\n<p>Answer:<br>\n\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-6808\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-4-Addition-of-Vectors-Ex-4b-5.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 4 Addition of Vectors Ex 4(b) 5\" width=\"328\" height=\"228\" sizes=\"auto, (max-width: 328px) 100vw, 328px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 4 Addition of Vectors Ex 4(b) 5\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 4 Addition of Vectors Ex 4(a)\"><br>\n\n<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-15-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-168\" style=\"width: 8.297em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 7.132em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.121em, 1007.13em, 2.658em, -999.998em); top: -2.328em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-169\"><span class=\"munderover\" id=\"MathJax-Span-170\"><span style=\"display: inline-block; position: relative; width: 1.447em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1001.4em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-171\"><span class=\"mrow\" id=\"MathJax-Span-172\"><span class=\"mi\" id=\"MathJax-Span-173\" style=\"font-family: MathJax_Main;\">A<\/span><span class=\"mi\" id=\"MathJax-Span-174\" style=\"font-family: MathJax_Main;\">B<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1001.45em, 3.777em, -999.998em); top: -4.518em; left: 0.002em;\"><span class=\"mo\" id=\"MathJax-Span-175\" style=\"\"><span style=\"display: inline-block; position: relative; width: 1.447em; height: 0px;\"><span style=\"position: absolute; top: -4.005em; left: -0.044em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 1.121em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.142em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.329em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.562em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.748em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.981em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-176\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">=<\/span><span class=\"munderover\" id=\"MathJax-Span-177\" style=\"padding-left: 0.282em;\"><span style=\"display: inline-block; position: relative; width: 0.562em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.51em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-178\"><span class=\"mrow\" id=\"MathJax-Span-179\"><span class=\"mi\" id=\"MathJax-Span-180\" style=\"font-family: MathJax_Main;\">b<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1000.51em, 3.777em, -999.998em); top: -4.518em; left: 0.002em;\"><span class=\"mo\" id=\"MathJax-Span-181\" style=\"\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; top: -4.005em; left: -0.044em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.235em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.096em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-182\" style=\"font-family: MathJax_Main;\">,<\/span><span class=\"munderover\" id=\"MathJax-Span-183\" style=\"padding-left: 0.189em;\"><span style=\"display: inline-block; position: relative; width: 1.493em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1001.4em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-184\"><span class=\"mrow\" id=\"MathJax-Span-185\"><span class=\"mi\" id=\"MathJax-Span-186\" style=\"font-family: MathJax_Main;\">A<\/span><span class=\"mi\" id=\"MathJax-Span-187\" style=\"font-family: MathJax_Main;\">C<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1001.45em, 3.777em, -999.998em); top: -4.518em; left: 0.002em;\"><span class=\"mo\" id=\"MathJax-Span-188\" style=\"\"><span style=\"display: inline-block; position: relative; width: 1.447em; height: 0px;\"><span style=\"position: absolute; top: -4.005em; left: -0.044em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 1.167em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.142em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.329em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.562em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.795em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.981em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-189\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">=<\/span><span class=\"munderover\" id=\"MathJax-Span-190\" style=\"padding-left: 0.282em;\"><span style=\"display: inline-block; position: relative; width: 0.468em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.42em, 4.15em, -999.998em); top: -4.005em; left: 0.002em;\"><span class=\"texatom\" id=\"MathJax-Span-191\"><span class=\"mrow\" id=\"MathJax-Span-192\"><span class=\"mi\" id=\"MathJax-Span-193\" style=\"font-family: MathJax_Main;\">c<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1000.47em, 3.777em, -999.998em); top: -4.285em; left: 0em;\"><span class=\"mo\" id=\"MathJax-Span-194\" style=\"\"><span style=\"display: inline-block; position: relative; width: 0.468em; height: 0px;\"><span style=\"position: absolute; top: -4.005em; left: -0.044em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.189em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.332em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.268em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-15\">\\overline{\\mathrm{AB}}=\\overline{\\mathrm{b}}, \\overline{\\mathrm{AC}}=\\overline{\\mathrm{c}}<\/script> and <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-16-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-195\" style=\"width: 3.963em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 3.404em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.121em, 1003.4em, 2.472em, -999.998em); top: -2.328em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-196\"><span class=\"munderover\" id=\"MathJax-Span-197\"><span style=\"display: inline-block; position: relative; width: 1.493em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1001.45em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-198\"><span class=\"mrow\" id=\"MathJax-Span-199\"><span class=\"mi\" id=\"MathJax-Span-200\" style=\"font-family: MathJax_Main;\">A<\/span><span class=\"mi\" id=\"MathJax-Span-201\" style=\"font-family: MathJax_Main;\">D<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1001.49em, 3.777em, -999.998em); top: -4.518em; left: 0.002em;\"><span class=\"mo\" id=\"MathJax-Span-202\" style=\"\"><span style=\"display: inline-block; position: relative; width: 1.493em; height: 0px;\"><span style=\"position: absolute; top: -4.005em; left: -0.044em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 1.167em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.096em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.282em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.468em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.655em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.841em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 1.027em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-203\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">=<\/span><span class=\"munderover\" id=\"MathJax-Span-204\" style=\"padding-left: 0.282em;\"><span style=\"display: inline-block; position: relative; width: 0.562em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.51em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-205\"><span class=\"mrow\" id=\"MathJax-Span-206\"><span class=\"mi\" id=\"MathJax-Span-207\" style=\"font-family: MathJax_Main;\">d<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1000.51em, 3.777em, -999.998em); top: -4.518em; left: 0.002em;\"><span class=\"mo\" id=\"MathJax-Span-208\" style=\"\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; top: -4.005em; left: -0.044em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.235em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.096em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.332em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.051em; border-left: 0px solid; width: 0px; height: 1.354em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-16\">\\overline{\\mathrm{AD}}=\\overline{\\mathrm{d}}<\/script><br> Let M, N be the mid points of one pair of opposite sides AB and CD of a quadrilateral ABCD.<br> <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-17-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-209\" style=\"width: 4.429em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 3.823em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.074em, 1003.82em, 2.845em, -999.998em); top: -2.328em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-210\"><span class=\"munderover\" id=\"MathJax-Span-211\"><span style=\"display: inline-block; position: relative; width: 1.68em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1001.63em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-212\"><span class=\"mrow\" id=\"MathJax-Span-213\"><span class=\"mi\" id=\"MathJax-Span-214\" style=\"font-family: MathJax_Main;\">A<\/span><span class=\"mi\" id=\"MathJax-Span-215\" style=\"font-family: MathJax_Main;\">M<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1001.63em, 3.777em, -999.998em); top: -4.518em; left: 0.002em;\"><span class=\"mo\" id=\"MathJax-Span-216\" style=\"\"><span style=\"display: inline-block; position: relative; width: 1.633em; height: 0px;\"><span style=\"position: absolute; top: -4.005em; left: -0.044em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 1.354em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.142em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.329em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.562em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.748em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.981em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 1.167em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-217\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">=<\/span><span class=\"mfrac\" id=\"MathJax-Span-218\" style=\"padding-left: 0.282em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.124em, 1000.38em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"munderover\" id=\"MathJax-Span-219\"><span style=\"display: inline-block; position: relative; width: 0.375em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.357em, 1000.38em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-220\"><span class=\"mrow\" id=\"MathJax-Span-221\"><span class=\"mi\" id=\"MathJax-Span-222\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">b<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.497em, 1000.38em, 3.87em, -999.998em); top: -4.378em; left: 0.002em;\"><span class=\"mo\" id=\"MathJax-Span-223\" style=\"\"><span style=\"display: inline-block; position: relative; width: 0.375em; height: 0px;\"><span style=\"position: absolute; top: -4.005em; left: -0.044em;\"><span style=\"font-size: 50%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.189em;\"><span style=\"font-size: 50%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.096em;\"><span style=\"font-size: 50%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-224\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.51em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.515em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.332em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.841em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-17\">\\overline{\\mathrm{AM}}=\\frac{\\overline{\\mathrm{b}}}{2}<\/script><br> <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-18-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-225\" style=\"width: 5.268em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 4.522em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.027em, 1004.52em, 2.845em, -999.998em); top: -2.328em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-226\"><span class=\"munderover\" id=\"MathJax-Span-227\"><span style=\"display: inline-block; position: relative; width: 1.493em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1001.49em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-228\"><span class=\"mrow\" id=\"MathJax-Span-229\"><span class=\"mi\" id=\"MathJax-Span-230\" style=\"font-family: MathJax_Main;\">A<\/span><span class=\"mi\" id=\"MathJax-Span-231\" style=\"font-family: MathJax_Main;\">N<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1001.49em, 3.777em, -999.998em); top: -4.518em; left: 0.002em;\"><span class=\"mo\" id=\"MathJax-Span-232\" style=\"\"><span style=\"display: inline-block; position: relative; width: 1.493em; height: 0px;\"><span style=\"position: absolute; top: -4.005em; left: -0.044em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 1.167em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.096em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.282em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.468em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.655em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.841em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 1.027em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-233\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">=<\/span><span class=\"mfrac\" id=\"MathJax-Span-234\" style=\"padding-left: 0.282em;\"><span style=\"display: inline-block; position: relative; width: 1.4em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.124em, 1001.26em, 4.196em, -999.998em); top: -4.471em; left: 50%; margin-left: -0.603em;\"><span class=\"mrow\" id=\"MathJax-Span-235\"><span class=\"munderover\" id=\"MathJax-Span-236\"><span style=\"display: inline-block; position: relative; width: 0.329em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.544em, 1000.28em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-237\"><span class=\"mrow\" id=\"MathJax-Span-238\"><span class=\"mi\" id=\"MathJax-Span-239\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">c<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.497em, 1000.33em, 3.87em, -999.998em); top: -4.192em; left: 0.002em;\"><span class=\"mo\" id=\"MathJax-Span-240\" style=\"\"><span style=\"display: inline-block; position: relative; width: 0.329em; height: 0px;\"><span style=\"position: absolute; top: -4.005em; left: -0.044em;\"><span style=\"font-size: 50%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.096em;\"><span style=\"font-size: 50%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.049em;\"><span style=\"font-size: 50%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-241\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">+<\/span><span class=\"munderover\" id=\"MathJax-Span-242\"><span style=\"display: inline-block; position: relative; width: 0.375em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.357em, 1000.38em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-243\"><span class=\"mrow\" id=\"MathJax-Span-244\"><span class=\"mi\" id=\"MathJax-Span-245\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">d<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.497em, 1000.38em, 3.87em, -999.998em); top: -4.378em; left: 0.002em;\"><span class=\"mo\" id=\"MathJax-Span-246\" style=\"\"><span style=\"display: inline-block; position: relative; width: 0.375em; height: 0px;\"><span style=\"position: absolute; top: -4.005em; left: -0.044em;\"><span style=\"font-size: 50%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.189em;\"><span style=\"font-size: 50%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.096em;\"><span style=\"font-size: 50%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-247\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1001.4em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.4em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.332em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.895em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-18\">\\overline{\\mathrm{AN}}=\\frac{\\overline{\\mathrm{c}}+\\overline{\\mathrm{d}}}{2}<\/script><br> Let P be the point of intersection of mid points of sides AB, CD and pair of diagonals AC, BD respectively.<br> Let <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-19-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-248\" style=\"width: 3.684em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 3.171em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.121em, 1003.12em, 2.472em, -999.998em); top: -2.328em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-249\"><span class=\"munderover\" id=\"MathJax-Span-250\"><span style=\"display: inline-block; position: relative; width: 1.447em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1001.35em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-251\"><span class=\"mrow\" id=\"MathJax-Span-252\"><span class=\"mi\" id=\"MathJax-Span-253\" style=\"font-family: MathJax_Main;\">A<\/span><span class=\"mi\" id=\"MathJax-Span-254\" style=\"font-family: MathJax_Main;\">P<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1001.4em, 3.777em, -999.998em); top: -4.518em; left: 0.002em;\"><span class=\"mo\" id=\"MathJax-Span-255\" style=\"\"><span style=\"display: inline-block; position: relative; width: 1.4em; height: 0px;\"><span style=\"position: absolute; top: -4.005em; left: -0.044em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 1.121em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.142em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.329em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.562em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.748em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.934em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-256\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">=<\/span><span class=\"munderover\" id=\"MathJax-Span-257\" style=\"padding-left: 0.282em;\"><span style=\"display: inline-block; position: relative; width: 0.375em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.451em, 1000.38em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-258\"><span class=\"mrow\" id=\"MathJax-Span-259\"><span class=\"mi\" id=\"MathJax-Span-260\" style=\"font-family: MathJax_Main;\">r<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1000.33em, 3.777em, -999.998em); top: -4.285em; left: 0.002em;\"><span class=\"mo\" id=\"MathJax-Span-261\" style=\"\"><span><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.332em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.051em; border-left: 0px solid; width: 0px; height: 1.354em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-19\">\\overline{\\mathrm{AP}}=\\overline{\\mathrm{r}}<\/script>. Then equation of <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-20-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-262\" style=\"width: 1.726em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.493em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.074em, 1001.49em, 2.425em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-263\"><span class=\"munderover\" id=\"MathJax-Span-264\"><span style=\"display: inline-block; position: relative; width: 1.493em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1001.4em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-265\"><span class=\"mrow\" id=\"MathJax-Span-266\"><span class=\"mi\" id=\"MathJax-Span-267\" style=\"font-family: MathJax_Main;\">A<\/span><span class=\"mi\" id=\"MathJax-Span-268\" style=\"font-family: MathJax_Main;\">C<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1001.45em, 3.777em, -999.998em); top: -4.518em; left: 0.002em;\"><span class=\"mo\" id=\"MathJax-Span-269\" style=\"\"><span style=\"display: inline-block; position: relative; width: 1.447em; height: 0px;\"><span style=\"position: absolute; top: -4.005em; left: -0.044em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 1.167em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.142em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.329em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.562em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.795em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.981em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.051em; border-left: 0px solid; width: 0px; height: 1.354em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-20\">\\overline{\\mathrm{AC}}<\/script> is<br> r\u0305 = t c\u0305 where t is a scalar \u2026\u2026\u2026\u2026.(1)<br> Equation of BD is<br> r\u0305 = (1 \u2013 s)b\u0305 + sd\u0305 for some scalars \u2026\u2026\u2026..(2)<br> and equation of line MN is<br> r\u0305 = (1 \u2013 \u03b1)<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-21-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-270\" style=\"width: 0.888em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 0.748em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.121em, 1000.75em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-271\"><span class=\"mfrac\" id=\"MathJax-Span-272\"><span style=\"display: inline-block; position: relative; width: 0.468em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.33em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"texatom\" id=\"MathJax-Span-273\"><span class=\"mrow\" id=\"MathJax-Span-274\"><span class=\"munderover\" id=\"MathJax-Span-275\"><span style=\"display: inline-block; position: relative; width: 0.375em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.357em, 1000.28em, 4.15em, -999.998em); top: -4.005em; left: 0.049em;\"><span class=\"mi\" id=\"MathJax-Span-276\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">b<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.451em, 1000.33em, 3.777em, -999.998em); top: -4.238em; left: 0em;\"><span class=\"mo\" id=\"MathJax-Span-277\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-278\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.47em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.468em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.732em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-21\">\\frac{\\bar{b}}{2}<\/script> + \u03b1<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-22-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-279\" style=\"width: 3.637em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 3.124em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.866em, 1002.94em, 3.963em, -999.998em); top: -3.166em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-280\"><span class=\"mrow\" id=\"MathJax-Span-281\"><span class=\"mo\" id=\"MathJax-Span-282\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size2;\">(<\/span><\/span><span class=\"mfrac\" id=\"MathJax-Span-283\"><span style=\"display: inline-block; position: relative; width: 1.633em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.218em, 1001.45em, 4.196em, -999.998em); top: -4.471em; left: 50%; margin-left: -0.743em;\"><span class=\"mrow\" id=\"MathJax-Span-284\"><span class=\"texatom\" id=\"MathJax-Span-285\"><span class=\"mrow\" id=\"MathJax-Span-286\"><span class=\"munderover\" id=\"MathJax-Span-287\"><span style=\"display: inline-block; position: relative; width: 0.422em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.544em, 1000.33em, 4.15em, -999.998em); top: -4.005em; left: 0.049em;\"><span class=\"mi\" id=\"MathJax-Span-288\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">c<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.451em, 1000.33em, 3.777em, -999.998em); top: -4.052em; left: 0.049em;\"><span class=\"mo\" id=\"MathJax-Span-289\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-290\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">+<\/span><span class=\"texatom\" id=\"MathJax-Span-291\"><span class=\"mrow\" id=\"MathJax-Span-292\"><span class=\"munderover\" id=\"MathJax-Span-293\"><span style=\"display: inline-block; position: relative; width: 0.562em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.357em, 1000.38em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-294\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">d<span style=\"display: inline-block; overflow: hidden; height: 1px; width: 0.002em;\"><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.451em, 1000.33em, 3.777em, -999.998em); top: -4.238em; left: 0.189em;\"><span class=\"mo\" id=\"MathJax-Span-295\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-296\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1001.63em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.633em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-297\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size2;\">)<\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 3.171em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.808em; border-left: 0px solid; width: 0px; height: 2.219em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-22\">\\left(\\frac{\\bar{c}+\\bar{d}}{2}\\right)<\/script><br> where \u03b1 is a scalar<br> \u21d2 2r\u0305 = (1 \u2013 \u03b1)b\u0305 + a(c\u0305 + d\u0305)<br> \u21d2 r\u0305 + r\u0305 = (1 \u2013 \u03b1)b\u0305 + a(c\u0305 + d\u0305)<br> From (1) and (2)<br> tc\u0305 + (1 \u2013 s)b\u0305 + sd\u0305 = (1 \u2013 \u03b1)b\u0305 + \u03b1(c\u0305 + d\u0305)<br> Equating coefficients of b, c, d we get<br> 1 \u2013 s = 1 \u2013 oc \u21d2 s = a and .<br> t = a \u21d2 s = t = a<br> From (1) and (2),<br> t c\u0305 = (1 \u2013 s) b\u0305 + s d\u0305<br> \u21d2 sc\u0305 = (1 \u2013 s) b\u0305 + s d\u0305 (\u2018- t = s)<br> \u21d2 (1 -s) b\u0305 =s(c\u0305 \u2013 d\u0305)<br> \u21d2 b is parallel to c\u0305 \u2013 d\u0305<br> \u21d2 AB is parallel to CD<br> \u2234 ABCD is a trapezium.<\/p>\n\n<h3>III.<br> Question 1.<br> Find the vector equation of the plane which passes through the points 2i\u0305 + 4j\u0305 + 2k\u0305, 2i\u0305 + 3j\u0305 + 5k\u0305 and parallel to the vector 3 i\u0305 \u2013 2 j\u0305 + k\u0305. Also find the point where this plane meets the line joining the points 2 i\u0305 + j\u0305 + 3k\u0305 and 4 i\u0305 \u2013 2 j\u0305 + 3k\u0305. (March 2012) (E.Q)<\/h3>\n\n<p>Answer:<br> Vector equation of the plane which passes through the points a\u0305 = 2i\u0305 + 4j\u0305 + 2k\u0305, b\u0305 = 2i\u0305 + 3j\u0305 + 5k\u0305 and parallel to vector c\u0305 = 3i\u0305 \u2013 2j\u0305 + k\u0305 is<br> r\u0305 = (1 \u2013 t)a\u0305 + tb\u0305 + sc\u0305 where t, s e R<br> \u21d2 r\u0305 = (1 \u2013 t) (2i\u0305 + 4j\u0305 + 2k\u0305) + t(2i\u0305 + 3j\u0305 + 5k\u0305) + s(3i\u0305 \u2013 2j\u0305 + k\u0305)<br> \u21d2 r\u0305 = (2 \u2013 2t + 2t + 3s) i\u0305 + (4 \u2013 4t + 3t \u2013 2s) j\u0305 + (2 \u2013 2t + 5t + s) k\u0305<br> \u21d2 r\u0305 = (2 \u2013 2t + 2t + 3s) i\u0305 + (4 \u2013 4t + 3t \u2013 2s) j\u0305 + (2 \u2013 2t + 5t + s) k\u0305<br> \u21d2 r\u0305 = (2 + 3s) i\u0305 + (4 \u2013 t \u2013 2s) j\u0305 + (2 + 3t + s)k\u0305 \u2026\u2026\u2026\u2026(1)<br> Vector equation of the line passing through the points c\u0305 = 2i\u0305 + j\u0305 + 3k\u0305 and d\u0305 = 4 i\u0305 \u2013 2 j\u0305 + 3k\u0305 is r\u0305 = (1 \u2013 a)d\u0305 + ac\u0305 where a e R<br> \u21d2 r\u0305 = (1 \u2013 \u03b1)(2i\u0305 + j\u0305 + 3k\u0305) + \u03b1(4i\u0305 \u2013 2j\u0305 + 3k\u0305)<br> \u21d2 r\u0305 = (2 \u2013 2\u03b1 + 4\u03b1) i\u0305 + (1 \u2013 \u03b1 \u2013 2\u03b1) j\u0305 + (3 \u2013 3\u03b1 + 3\u03b1)k\u0305<br> \u21d2 r\u0305 = (2 + 2\u03b1) i\u0305 + (1 \u2013 3\u03b1) j\u0305 + 3k\u0305 (2)<br> Let 7 be the point of intersection of (1) and (2)<br> (2 + 3s)i\u0305 + (4 \u2013 t \u2013 2s) j\u0305 + (2 + 3t + s) k\u0305<br> = (2 + 2\u03b1) i\u0305 + (1 \u2013 3\u03b1) j\u0305 + 3k\u0305<br> v Since i\u0305, j\u0305, k\u0305 are non coplanar,<br> 2 + 3s = 2 + 2\u03b1 \u21d2 2\u03b1 \u2013 3s = 0 \u2026\u2026\u2026\u2026\u2026\u2026(3)<br> 4 \u2013 1 \u2013 2s = 1 \u2013 3\u03b1 \u21d2 3\u03b1 \u2013 2s -1 = \u2013 3 \u2026\u2026\u2026\u2026(4)<br> 2 + 3t + s = 3 \u21d2 s + 3t = 1 \u2026\u2026\u2026\u2026\u2026\u2026(5)<br> From (5), t = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-23-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-298\" style=\"width: 1.866em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.587em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.214em, 1001.59em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-299\"><span class=\"mfrac\" id=\"MathJax-Span-300\"><span style=\"display: inline-block; position: relative; width: 1.307em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1001.17em, 4.196em, -999.998em); top: -4.471em; left: 50%; margin-left: -0.603em;\"><span class=\"mrow\" id=\"MathJax-Span-301\"><span class=\"mn\" id=\"MathJax-Span-302\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span class=\"mo\" id=\"MathJax-Span-303\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2212<\/span><span class=\"texatom\" id=\"MathJax-Span-304\"><span class=\"mrow\" id=\"MathJax-Span-305\"><span class=\"mi\" id=\"MathJax-Span-306\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">s<\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-307\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1001.31em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.307em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.624em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-23\">\\frac{1-\\mathrm{s}}{3}<\/script><br> \u2234 From (4) 3\u03b1 \u2013 2s \u2013 <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-24-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-308\" style=\"width: 3.311em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 2.845em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.866em, 1002.66em, 3.963em, -999.998em); top: -3.166em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-309\"><span class=\"mrow\" id=\"MathJax-Span-310\"><span class=\"mo\" id=\"MathJax-Span-311\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size2;\">(<\/span><\/span><span class=\"mfrac\" id=\"MathJax-Span-312\"><span style=\"display: inline-block; position: relative; width: 1.354em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1001.21em, 4.196em, -999.998em); top: -4.471em; left: 50%; margin-left: -0.603em;\"><span class=\"mrow\" id=\"MathJax-Span-313\"><span class=\"mn\" id=\"MathJax-Span-314\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span class=\"mo\" id=\"MathJax-Span-315\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2212<\/span><span class=\"mi\" id=\"MathJax-Span-316\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">s<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-317\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1001.35em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.354em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-318\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size2;\">)<\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 3.171em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.808em; border-left: 0px solid; width: 0px; height: 2.219em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-24\">\\left(\\frac{1-s}{3}\\right)<\/script> = -3<br> \u21d2 9\u03b1 \u2013 6s \u2013 1 + s = -9<br> 9\u03b1 \u2013 5s + 8 = 0 (6)<br> Solving (6) &amp; (3) equations<br>\n\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-6807\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-4-Addition-of-Vectors-Ex-4b-6.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 4 Addition of Vectors Ex 4(b) 6\" width=\"324\" height=\"458\" sizes=\"auto, (max-width: 324px) 100vw, 324px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 4 Addition of Vectors Ex 4(b) 6\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 4 Addition of Vectors Ex 4(a)\"><\/p>\n\n<h3>Question 2.<br> Find the vector equation of the plane passing through the points 4 i\u0305 \u2013 3 j\u0305 \u2013 k\u0305 , 3i\u0305 + 7j\u0305 \u2013 10k\u0305 and 2i\u0305 + 5j\u0305 \u2013 7k\u0305 and show that the point i\u0305 + 2 j\u0305 \u2013 3k\u0305 lies in the plane. (March 2013) (S.A.Q.)<\/h3>\n\n<p>Answer:<br> Vector equation of the plane passing through<br> A(4i\u0305 \u2013 3j\u0305 \u2013 k\u0305 ), B (3i\u0305 + 7j\u0305 \u2013 10k\u0305 ) and C(2i\u0305 + 5j\u0305 \u2013 7k\u0305 ) is<br> r\u0305 = (1 \u2013 s \u2013 t) (4i\u0305 \u2013 3 j\u0305 \u2013 k\u0305 ) + s(3i\u0305 + 7j\u0305 \u2013 10k\u0305 ) + t(2i\u0305 + 5j\u0305 \u2013 7k\u0305 )<br> Let D (i\u0305 + 2j\u0305 \u2013 3k\u0305 ) lies on the plane, then<br> (i\u0305 + 2j\u0305 \u2013 3k\u0305 ) = (1 \u2013 s \u2013 t)(4i\u0305 \u2013 3j\u0305 \u2013 k\u0305 ) + s (3i\u0305 + 7j\u0305 \u2013 10k\u0305 ) + t (2i\u0305 + 5j\u0305 \u2013 7k\u0305 )<br> Since i\u0305 , j\u0305 ,k\u0305 are non coplanar, equating coefficients of i\u0305 , j\u0305 , k\u0305 both sides.<br> 4(1 \u2013 s \u2013 t) + 3s + 2t = 1<br> \u21d2 4 \u2013 4s \u2013 4t + 3s + 2t = 1<br> \u21d2 s + 2t = 3 \u2026\u2026\u2026\u2026(1)<br> \u2013 3 (1 \u2013 s \u2013 t) + 7s + 5t = 2<br> \u21d2 -3 + 3s + 3t + 7s + 5t = 2<br> \u21d2 10s + 8t = 5<br> Also \u2013 (1 \u2013 s \u2013 t) \u2013 10s \u2013 7t = \u2013 3<br> \u21d2 \u2013 1 + s + t \u2013 10s \u2013 7t = \u2013 3<br> \u21d2 9s + 6t = 2<br> From (1), 3s + 6t = 9<br> Solving (1) &amp; (3) equations 6s = \u2013 7<br> \u21d2 s = \u2013 7\/6<br>\n\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-6806\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-4-Addition-of-Vectors-Ex-4b-7.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 4 Addition of Vectors Ex 4(b) 7\" width=\"267\" height=\"285\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 4 Addition of Vectors Ex 4(b) 7\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 4 Addition of Vectors Ex 4(a)\"><br>\n\ns = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-25-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-319\" style=\"width: 1.54em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.307em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.214em, 1001.31em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-320\"><span class=\"mfrac\" id=\"MathJax-Span-321\"><span style=\"display: inline-block; position: relative; width: 1.027em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.89em, 4.196em, -999.998em); top: -4.471em; left: 50%; margin-left: -0.464em;\"><span class=\"mrow\" id=\"MathJax-Span-322\"><span class=\"mo\" id=\"MathJax-Span-323\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-324\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">7<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-325\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">6<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1001.03em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.027em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.624em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-25\">\\frac{-7}{6}<\/script> t = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-26-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-326\" style=\"width: 1.307em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.121em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.26em, 1001.12em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-327\"><span class=\"mfrac\" id=\"MathJax-Span-328\"><span style=\"display: inline-block; position: relative; width: 0.841em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.65em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.37em;\"><span class=\"mn\" id=\"MathJax-Span-329\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">25<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.65em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.37em;\"><span class=\"mn\" id=\"MathJax-Span-330\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">12<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.84em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.841em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-26\">\\frac{25}{12}<\/script>. satisfy (1), (2), (3).<br> and D lies on the plane passing through A, B, C.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>I. Question 1. Find the vector equation of the line passing through the point 2i\u0305 + 3j\u0305 + k\u0305 and parallel to the vector 4i\u0305 \u2013 2j\u0305 + 3k\u0305.(March 2015-A.P) (May, March \u201901) (V.S.A) Answer: Let a = 2i\u0305 + 3j\u0305 + k\u0305 and b = 4i\u0305 \u2013 2 j\u0305 + 3k\u0305 The vector equation [&hellip;]<\/p>\n","protected":false},"author":2,"featured_media":2932,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[1],"tags":[],"class_list":{"0":"post-2909","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-other"},"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\r\n<title>TS Inter 1st Year Maths 1A Addition of Vectors Solutions Exercise 4(b)<\/title>\r\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\r\n<link rel=\"canonical\" href=\"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-addition-of-vectors-solutions-exercise-4b\/\" \/>\r\n<meta property=\"og:locale\" content=\"en_US\" \/>\r\n<meta property=\"og:type\" content=\"article\" \/>\r\n<meta property=\"og:title\" content=\"TS Inter 1st Year Maths 1A Addition of Vectors Solutions Exercise 4(b)\" \/>\r\n<meta property=\"og:description\" content=\"I. Question 1. Find the vector equation of the line passing through the point 2i\u0305 + 3j\u0305 + k\u0305 and parallel to the vector 4i\u0305 \u2013 2j\u0305 + 3k\u0305.(March 2015-A.P) (May, March \u201901) (V.S.A) Answer: Let a = 2i\u0305 + 3j\u0305 + k\u0305 and b = 4i\u0305 \u2013 2 j\u0305 + 3k\u0305 The vector equation [&hellip;]\" \/>\r\n<meta property=\"og:url\" content=\"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-addition-of-vectors-solutions-exercise-4b\/\" \/>\r\n<meta property=\"og:site_name\" content=\"Manabadi Boards\" \/>\r\n<meta property=\"article:published_time\" content=\"2026-01-19T10:28:42+00:00\" \/>\r\n<meta property=\"article:modified_time\" content=\"2026-01-19T10:28:44+00:00\" \/>\r\n<meta property=\"og:image\" content=\"https:\/\/boardscdn.manabadi.co.in\/wp-content\/uploads\/2026\/01\/07114619\/ts-inter-1st-year-maths-1a-solutions-chp-4-addition-of-vectors-ex-4b.jpg\" \/>\r\n\t<meta property=\"og:image:width\" content=\"900\" \/>\r\n\t<meta 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