{"id":14784,"date":"2021-05-12T11:37:25","date_gmt":"2021-05-12T06:07:25","guid":{"rendered":"https:\/\/yanamtakshashila.com\/?p=14784"},"modified":"2023-04-30T20:10:51","modified_gmt":"2023-04-30T14:40:51","slug":"1-3-n-conics-exercise-problems-with-solutions","status":"publish","type":"post","link":"https:\/\/yanamtakshashila.com\/?p=14784","title":{"rendered":"1.2 &#8211; N &#8211; CONICS &#8211; Exercise Problems with solutions"},"content":{"rendered":"\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\LARGE{\\color {purple} {PART- A}}\\]<script src=\"https:\/\/yanamtakshashila.com\/wp-includes\/js\/dist\/hooks.min.js?ver=dd5603f07f9220ed27f1\" id=\"wp-hooks-js\"><\/script>\n<script src=\"https:\/\/yanamtakshashila.com\/wp-includes\/js\/dist\/i18n.min.js?ver=c26c3dc7bed366793375\" id=\"wp-i18n-js\"><\/script>\n<script id=\"wp-i18n-js-after\">\nwp.i18n.setLocaleData( { 'text direction\\u0004ltr': [ 'ltr' ] } );\n\/\/# sourceURL=wp-i18n-js-after\n<\/script>\n<script  async src=\"https:\/\/cdnjs.cloudflare.com\/ajax\/libs\/mathjax\/2.7.7\/MathJax.js?config=TeX-MML-AM_CHTML\" id=\"mathjax-js\"><\/script>\n<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {1\\ .}\\ \\color {red} {Define}\\ Conic\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\  A\\ conic\\ is\\ defined\\ as\\ the\\ locus\\ of\\ a\\ point\\ which\\ moves\\ such\\ that\\ its\\ distance\\ from\\ a\\ fixed\\ point\\ is\\ \\hspace{5cm}\\]\\[always \u2018e\u2019\\ times\\ its\\ distance\\ from\\ a\\ fixed\\ straight line.\\ \\hspace{10cm}\\]     <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {2\\ .}\\ \\color {red} {Prove\\ that}\\ the\\ equation\\ x^2\\ -\\ 2\\ x\\ y\\ +\\  y^2\\ -\\ 16\\ x\\ -\\ 12\\ y\\ +\\ 22\\ = 0\\ is\\ a\\ parabola\\ \\hspace{5cm}\\]  <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ x^2\\ -\\ 2\\ x\\ y\\ +\\ y^2\\ -\\ 16\\ x\\ -\\ 12\\ y\\ +\\ 22\\ =\\ 0\\ &#8212;&#8212;&#8212;- (1)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ Condition\\ for\\ (1)\\ to\\ represent\\ parabola\\ is\\ h^2\\ =\\ ab\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Comparing\\ with\\ a\\ x^2\\ +\\ 2h\\ x\\ y\\ +\\   by^2\\ +\\  2\\ g\\ x\\ +\\ 2\\ f\\ y\\ +\\ c\\ = 0\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[We\\ get\\  a\\ =\\ 1,\\ b\\ =\\ 1\\  \\hspace{5cm}\\ 2h\\ =\\ -\\ 2,\\ \\implies\\ h\\ =\\ -\\ 1\\  \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[h^2\\ =\\ ab\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(-1)^2\\ =\\ 1(1)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[1\\ =\\ 1\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\therefore\\ (1)\\ represents\\ a\\ parabola\\ \\hspace{10cm}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/LbrQQuj0iaU\" title=\"CONICS (Exercise) - Part - 2\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"aicp\"><script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js\"><\/script>\n<!-- sidebar ad 1 -->\n<ins class=\"adsbygoogle\" style=\"display:inline-block;width:326px;height:280px\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"6703350399\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {3\\ .}\\ \\color {red} {Show\\ that}\\ the\\ equation\\  7x^2\\ +\\ 3xy\\ +\\ 2y^2\\ -\\ \\ x\\ +\\ 2\\ y\\ -\\ 1\\ =\\ 0\\ \\hspace{10cm}\\]\\[\\color {red} {represents\\ an\\ ellipse}\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ 7x^2\\ +\\ 3xy\\ +\\ 2y^2\\ -\\ \\ x\\ +\\ 2\\ y\\ -\\ 1\\ =\\ 0\\ &#8212;&#8212;&#8212;- (1)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ Condition\\ for\\ (1)\\ to\\ represent\\ an\\ ellipse\\ is\\ h^2\\ -\\ ab\\ \\lt\\  0\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Comparing\\ with\\ a\\ x^2\\ +\\ 2h\\ x\\ y\\ +\\   by^2\\ +\\  2\\ g\\ x\\ +\\ 2\\ f\\ y\\ +\\ c\\ = 0\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[We\\ get\\  a\\ =\\ 7,\\ b\\ =\\ 2\\  \\hspace{5cm}\\ 2h\\ =\\ 3,\\ \\implies\\ h\\ =\\ \\frac{3}{2}\\  \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ h^2\\ -\\ ab\\ =\\ (\\frac{3}{2})^2\\ -\\ 7(2)\\  =\\ \\frac{9}{4}\\ -\\ 14\\ =\\ -\\ \\frac{9\\ -\\ 56}{14}\\ =\\ -\\ \\frac{47}{14}\\ \\lt\\  0\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\therefore\\ (1)\\ represents\\ an\\ ellipse\\ \\hspace{10cm}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/7KRB_uRT6z0\" title=\"CONICS (Exercise) - PART - 3\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"aicp\"><script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js\"><\/script>\n<ins class=\"adsbygoogle\" style=\"display:block; text-align:center;\" data-ad-layout=\"in-article\" data-ad-format=\"fluid\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"2812384453\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\LARGE{\\color {purple} {PART- B}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {4\\ .}\\ \\color {red} {Show\\ that}\\ the\\ equation\\ 4\\ x^2\\ +\\; 10\\ x\\ y\\ +\\   y^2\\ -\\  2\\ x\\ +\\ 5\\ y\\ -\\ 3\\ = 0\\ \\color {red} {represents\\ a\\ hyperbola}\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ 4\\ x^2\\ +\\; 10\\ x\\ y\\ +\\   y^2\\ -\\  2\\ x\\ +\\ 5\\ y\\ -\\ 3\\ = 0\\ &#8212;&#8212;&#8212;- (1)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ Condition\\ for\\ (1)\\ to\\ represent\\ hyperbola\\ is\\ h^2\\ -\\ ab\\ \\gt\\  0\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Comparing\\ with\\ a\\ x^2\\ +\\ 2h\\ x\\ y\\ +\\   by^2\\ +\\  2\\ g\\ x\\ +\\ 2\\ f\\ y\\ +\\ c\\ = 0\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[We\\ get\\  a\\ =\\ 4,\\ b\\ =\\ 1\\  \\hspace{5cm}\\ 2h\\ =\\ 10,\\ \\implies\\ h\\ =\\ 5\\  \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ h^2\\ -\\ ab\\ =\\ (5)^2\\ -\\ 4(1)\\ =\\ 25\\ -\\ 4\\ =\\ 21\\ \\gt\\  0\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\therefore\\ (1)\\ represents\\ a\\ hyperbola\\ \\hspace{10cm}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/_b7EPJOpOTo\" title=\"CONICS (Exercise) - PART - 4\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {5\\ .}\\ \\color {red} {Find\\ the\\ equation\\ of\\ the\\ parabola}\\ with\\ focus\\ at\\ (-1,\\ -2)\\ and\\  x\\ +\\ 2\\ y\\ =\\ 0\\ as\\ its\\ directrix\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ For\\ parabola\\ e\\ =\\ 1\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ Given\\ Focus\\ is\\ S(-1,\\ -2)\\ and\\ directrix\\ is\\ x\\ +\\ 2\\ y\\ =\\ 0.\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Always\\ \\frac{SP}{PM}\\ =\\ e\\ =\\ 1\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\\sqrt{(x\\ -\\ x_1)^2\\ +\\ (y\\ -\\ y_1)^2}}{\\pm\\ \\frac{a\\ x\\ +\\ b\\ y\\ +\\ c}{\\sqrt{a^2\\ +\\ b^2}}}\\ =\\ 1\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\\sqrt{(x\\ +\\ 1)^2\\ +\\ (y\\ +\\ 2)^2}}{\\pm\\ \\frac{x\\ +\\ 2\\ y}{\\sqrt{(1)^2\\ +\\ (2)^2}}}\\ =\\ 1\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\sqrt{(x\\ +\\ 1)^2\\ +\\ (y\\ +\\ 2)^2}\\ =\\ \\pm\\ \\frac{x\\ +\\ 2\\ y}{\\sqrt{5}}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(x\\ +\\ 1)^2\\ +\\ (y\\ +\\ 2)^2\\ =\\ \\frac{(x\\ +\\ 2\\ y)^2}{5}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ 5(x^2\\ +\\ 2\\ x\\ +\\ 1\\ +\\ y^2\\ +\\ 4\\ y\\ +\\ 4)\\ =\\ x^2\\ +\\ 4\\ y^2\\ +\\ 4\\ x\\ y\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ 5\\ x^2\\ +\\ 10\\ x\\ +\\ 5\\ +\\ 5\\ y^2\\ +\\ 20\\ y\\ +\\ 20\\ -\\ x^2\\ -\\ 4\\ y^2\\ -\\ 4\\ x\\ y\\ =\\ 0\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ 4\\ x^2\\ -\\ 4\\ x\\ y\\ +\\ 10\\ x\\  +\\  y^2\\ +\\ 20\\ y\\ +\\ 25\\ =\\ 0\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ \\boxed {4\\ x^2\\ -\\ 4\\ x\\ y\\ +\\ 10\\ x\\  +\\  y^2\\ +\\ 20\\ y\\ +\\ 25\\ =\\ 0}\\ \\hspace{8cm}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/iAl4hKbCthM\" title=\"CONICS (Exercise) - PART - 05\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {6\\ .}\\ \\color{red}{What\\ is\\ the\\ condition\\ for\\ the\\ conic}\\ a\\ x^2\\ +\\ 2\\ h\\ x\\ y\\ +\\ b\\ y^2\\ +\\ 2\\ g\\ x\\ +\\ 2\\ f\\ y\\ +\\ c\\ =\\ 0\\ \\hspace{7cm}\\]\\[\\color {red} {represents\\ a\\ pair\\ of\\ straight\\ lines}\\  \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ The\\ condition\\ for\\ the\\ given\\ conic\\ to\\ represent\\ a\\ pair\\ of\\ straight\\ lines\\ is\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ a\\ b\\ c\\ +\\ 2\\ f\\ g\\ h\\ -\\ a\\ f^2\\ -\\ b\\ g^2\\ -\\ c\\ h^2\\ =\\ 0\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<p>    <\/p>\n\n\n\n<div class=\"aicp\"><script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js\"><\/script>\n<!-- Ad1 -->\n<ins class=\"adsbygoogle\" style=\"display:block\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"8240817448\" data-ad-format=\"auto\" data-full-width-responsive=\"true\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\LARGE{\\color {purple} {PART- C}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {7\\ .}\\ \\color{red}{Find\\ the\\ equation\\ of\\ the\\ Ellipse}\\ with\\ focus\\ (-1,\\ -3)\\ and\\ directrix\\ x\\ -\\ 2y\\ =\\ 0\\ and\\ e\\ =\\ \\frac{4}{5}\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ Given\\ Focus\\ is\\ S(-1,\\ -3)\\ and\\ directrix\\ is\\ x\\ -\\ 2y\\ =\\ 0,\\ e\\ =\\ \\frac{4}{5}\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\\sqrt{(x\\ -\\ x_1)^2\\ +\\ (y\\ -\\ y_1)^2}}{\\pm\\ \\frac{a\\ x\\ +\\ b\\ y\\ +\\ c}{\\sqrt{a^2\\ +\\ b^2}}}\\ =\\ e\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\\sqrt{(x\\ +\\ 1)^2\\ +\\ (y\\ +\\ 3)^2}}{\\pm\\ \\frac{x\\ -\\ 2y}{\\sqrt{(1)^2\\ +\\ (-2)^2}}}\\ =\\ \\frac{4}{5}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[5\\ \\sqrt{(x\\ +\\ 1)^2\\ +\\ (y\\ +\\ 3)^2}\\ =\\ 4\\  (\\pm\\ \\frac{x\\ -\\ 2y}{\\sqrt{5}})\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[25\\ ((x\\ +\\ 1)^2\\ +\\ (y\\ +\\ 3)^2)\\ =\\ 16\\ (\\frac{(x\\ -\\ 2y)^2}{5})\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ 125(x^2\\ +\\ 2\\ x\\ +\\ 1\\ +\\ y^2\\ +\\ 6\\ y\\ +\\ 9)\\ =\\ 16(x^2\\ -\\ 4\\ x\\ y\\ +\\ 4\\ y^2)\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ 125\\ x^2\\ +\\ 250\\ x\\ +\\ 125\\ +\\ 125\\ y^2\\ +\\  750\\ y\\ +\\ 1125\\ -\\ 16\\ x^2\\ +\\ 64\\ x\\ y\\  +\\ 64\\ =\\ 0\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ 109\\ x^2\\ +\\ 64\\ x\\ y\\ +\\ 250\\ x\\  +\\  61\\ y^2\\ +\\ 750\\ y\\ +\\ 1250\\ =\\ 0\\ \\hspace{8cm}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/2B2l5nTNiEo\" title=\"CONICS (Exercise) - PART - 7\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {8\\ .}\\ \\color{red}{Prove\\ that}\\ equation\\ 3\\ x^2\\ +\\ 7\\ x\\ y\\ +\\ 2\\ y^2\\ +\\ 5\\ x\\ +\\ 5\\ y\\ +\\ 2\\ =\\ 0\\ \\hspace{7cm}\\]\\[\\color {red} {represents\\ a\\ pair\\ of\\ straight\\ lines}\\  \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ Given\\ 3\\ x^2\\ +\\ 7\\ x\\ y\\ +\\ 2\\ y^2\\ +\\ 5\\ x\\ +\\ 5\\ y\\ +\\ 2\\ =\\ 0\\ &#8212;&#8212;&#8212;-(1)\\  \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ a\\ x^2\\ +\\ 2\\ h\\ x\\ y\\ +\\ b\\ y^2\\ +\\ 2\\ g\\ x\\ +\\ 2\\ f\\ y\\ +\\ c\\ =\\ 0\\  \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ comparing\\ we\\ get\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\  a\\ =\\ 3\\ \\hspace{2cm}\\ 2\\ h\\ =\\   7\\  \\hspace{2cm}\\ b\\ =\\ 2\\ \\hspace{2cm}\\ 2\\ g\\ =\\ 5\\ \\hspace{2cm}\\ 2\\ f\\ =\\ \\ 5\\ \\hspace{2cm}\\  c\\ =\\ 2\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{6cm}\\  h\\ =\\  \\frac{7}{2}\\  \\hspace{4cm}\\  g\\ =\\ \\frac{5}{2}\\ \\hspace{4cm}\\  f\\ =\\ \\frac{5}{2}\\ \\hspace{4cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ To\\ claim\\ equation\\ (1)\\ represents\\ a\\ pair\\ of\\ straight\\ lines\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ \\therefore\\ i.e\\ a\\ b\\ c\\ +\\ 2\\ f\\ g\\ h\\ -\\ a\\ f^2\\ -\\ b\\ g^2\\ -\\ c\\ h^2\\ =\\ 0\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ L.H.S\\ =\\ a\\ b\\ c\\ +\\ 2\\ f\\ g\\ h\\ -\\ a\\ f^2\\ -\\ b\\ g^2\\ -\\ c\\ h^2\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ =\\ 3\\ (2)\\ (2)\\ +\\ 2\\ (\\frac{5}{2})\\ (\\frac{5}{2})\\ (\\frac{7}{2})\\ -\\ 3\\ (\\frac{-5}{2}) ^2\\ -\\ 2\\ (\\frac{5}{2}) ^2\\ -\\ 2\\ (\\frac{7}{2}) ^2\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ =\\ 12\\ +\\ \\frac{175}{4}\\ -\\ 3\\ (\\frac{25}{4})\\ -\\ 2\\ (\\frac{25}{4})\\  -\\  2\\ (\\frac{49}{4})\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ 12\\ +\\ \\frac{175}{4}\\ -\\  \\frac{75}{4}\\ -\\  \\frac{50}{4}\\ -\\  \\frac{98}{4}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ 12\\ +\\  (\\frac{175\\ -\\ 75\\ -\\ 50\\ -\\ 98}{4})\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ 12\\ +\\  (\\frac{175\\ -\\ 223}{4})\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ 12\\ +\\  \\frac{- 48}{4}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ 12\\ -\\  12\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ 0\\ =\\ R.\\ H.\\ S\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ \\therefore\\ the\\ given\\ equation\\ (1)\\ represents\\ a\\ pair\\ of\\ straighlt\\ lines\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/TBgRS0rPw9A\" title=\"CONICS (Exercise) - PART -08\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe>\n\n\n\n<script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js?client=ca-pub-9453835310745500\" crossorigin=\"anonymous\"><\/script>\n<!-- wide skyscraper -->\n<ins class=\"adsbygoogle\" style=\"display:inline-block;width:300px;height:600px\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"9987820756\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/4rCQlvKE2TM\" title=\"Conics (Exercise) - Part - 9\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {10\\ .}\\ For\\ the\\ quadratic\\ equation\\ 2\\ x^2\\ +\\ 7\\ x\\ y\\ +\\ 3\\ y^2\\ +\\ 13\\ x\\ -\\ \\ y\\ -\\ 24\\ =\\ 0\\ \\hspace{7cm}\\]\\[\\color {red} {identify\\ the\\ conic}\\  \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ Given\\ 2\\ x^2\\ +\\ 7\\ x\\ y\\ +\\ 3\\ y^2\\ +\\ 13\\ x\\ -\\ \\ y\\ -\\ 24\\ =\\ 0\\ &#8212;&#8212;&#8212;-(1)\\  \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ a\\ x^2\\ +\\ 2\\ h\\ x\\ y\\ +\\ b\\ y^2\\ +\\ 2\\ g\\ x\\ +\\ 2\\ f\\ y\\ +\\ c\\ =\\ 0\\  \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ comparing\\ we\\ get\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\  a\\ =\\ 2\\ \\hspace{2cm}\\ 2\\ h\\ =\\  7\\  \\hspace{2cm}\\ b\\ =\\ 3\\ \\hspace{2cm}\\ 2\\ g\\ =\\ 13\\ \\hspace{2cm}\\ 2\\ f\\ =\\ &#8211; 1\\ \\hspace{2cm}\\  c\\ =\\ &#8211; 24\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{6cm}\\  h\\ =\\  \\frac{7}{2}\\  \\hspace{4cm}\\  g\\ =\\ \\frac{13}{2}\\ \\hspace{4cm}\\  f\\ =\\ \\frac{-1}{2}\\ \\hspace{4cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ To\\ claim\\ equation\\ (1)\\ represents\\ a\\ pair\\ of\\ straight\\ lines\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ \\therefore\\ i.e\\ a\\ b\\ c\\ +\\ 2\\ f\\ g\\ h\\ -\\ a\\ f^2\\ -\\ b\\ g^2\\ -\\ c\\ h^2\\ =\\ 0\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ L.H.S\\ =\\ a\\ b\\ c\\ +\\ 2\\ f\\ g\\ h\\ -\\ a\\ f^2\\ -\\ b\\ g^2\\ -\\ c\\ h^2\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ =\\ 2\\ (3)\\ (-24)\\ +\\ 2\\ (\\frac{-1}{2})\\  (\\frac{13}{2})\\ (\\frac{7}{2})\\ -\\ 2\\ (\\frac{- 1}{2}) ^2\\ -\\ 3(\\frac{13}{2})^2\\ +\\ 24\\ (\\frac{7}{2}) ^2\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ =\\ -\\ 144\\ -\\ \\frac{91}{4}\\ -\\ 2\\ (\\frac{1}{4})\\ -\\ 3\\ (\\frac{169}{4})\\ +\\  24\\ (\\frac{49}{4})\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ -\\ 144\\  -\\  \\frac{91}{4}\\ -\\ \\frac{2}{4}\\ -\\  \\frac{507}{4}\\ +\\ 294\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ 150\\ -\\  (\\frac{91\\ +\\ 2\\ +\\ 507}{4})\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ 150\\ -\\  \\frac{600}{4}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ 150\\ -\\  150\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ 0\\ =\\ R.\\ H.\\ S\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ \\therefore\\ the\\ given\\ equation\\ (1)\\ represents\\ a\\ pair\\ of\\ straighlt\\ lines\\ \\hspace{8cm}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/a8o0xgwJ2gs\" title=\"CONICS (Exercise) - PART - 09\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js?client=ca-pub-9453835310745500\" crossorigin=\"anonymous\"><\/script>\n<ins class=\"adsbygoogle\" style=\"display:block\" data-ad-format=\"autorelaxed\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"4869133702\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<p>                 <\/p>\n\n\n\n<p>                   <\/p>\n\n\n\n<p>                                    <\/p>\n\n\n\n<p>                   <\/p>\n\n\n\n<p>                    <\/p>\n\n\n\n<p>                     <\/p>\n\n\n\n<p>                               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