{"id":44805,"date":"2023-12-21T19:32:28","date_gmt":"2023-12-21T14:02:28","guid":{"rendered":"https:\/\/yanamtakshashila.com\/?p=44805"},"modified":"2026-02-11T20:21:05","modified_gmt":"2026-02-11T14:51:05","slug":"coordinate-geometry-non-circuit-and-circuity","status":"publish","type":"post","link":"https:\/\/yanamtakshashila.com\/?p=44805","title":{"rendered":"COORDINATE GEOMETRY (UNIT &#8211; 1 FOR CIRCUIT AND UNIT &#8211; 2 FOR NON &#8211; CIRCUIT)"},"content":{"rendered":"\n<h5 class=\"wp-block-heading has-vivid-purple-color has-text-color has-link-color wp-elements-c22ec3607bb6321651489bcfd9999123\">Syllabus:<\/h5>\n\n\n\n<p>General equation of conics&nbsp;\u2013&nbsp;Classification of conics&nbsp;\u2013&nbsp;Standard equations of parabola&nbsp;\u2013&nbsp;Vertex, focus, axis, directrix, focal distance, focal chord, latus-rectum of parabola&nbsp;,  Standard equations of ellipse&nbsp;\u2013&nbsp;Vertices, foci, major axis, minor axis, directrices, eccentricity, centre and latus-rectums of ellipse&nbsp;\u2013&nbsp;Simple problems.<\/p>\n\n\n\n<figure class=\"wp-block-image alignfull size-full\"><img data-recalc-dims=\"1\" decoding=\"async\" width=\"258\" height=\"185\" data-attachment-id=\"54593\" data-permalink=\"https:\/\/yanamtakshashila.com\/?attachment_id=54593\" data-orig-file=\"https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2023\/12\/conic.png?fit=258%2C185&amp;ssl=1\" data-orig-size=\"258,185\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"conic\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2023\/12\/conic.png?fit=258%2C185&amp;ssl=1\" src=\"https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2023\/12\/conic.png?resize=258%2C185&#038;ssl=1\" alt=\"\" class=\"wp-image-54593\"\/><\/figure>\n\n\n\n<p><strong>Conic: <\/strong>A conic is defined as the locus of a point which moves such that its distance from a fixed point is always \u2018e\u2019  times its distance from a fixed straight line.<\/p>\n\n\n\n<p><strong>Focus: <\/strong>The fixed point is called the focus of the conic.<\/p>\n\n\n\n<p><strong>Directrix: <\/strong>The fixed straight line is called the directrix of the conic.<\/p>\n\n\n\n<p><strong>Eccentricity: <\/strong>The constant ratio is called the eccentricity of the conic.<\/p>\n\n\n\n<p>S\u2019&nbsp; denotes Focus&nbsp;&nbsp;<\/p>\n\n\n\n<p>Line XM denotes Directrix<\/p>\n\n\n\n<p>SP \/ PM &nbsp;= e&nbsp;&nbsp;<\/p>\n\n\n\n<p>Note:<\/p>\n\n\n\n<p>(i) If&nbsp; e&lt; 1,&nbsp; the conic is called an ellipse.<\/p>\n\n\n\n<p>(ii) If&nbsp; e = 1,&nbsp; the conic is called a parabola.<\/p>\n\n\n\n<p>(iii) If e&gt;1, the conic is called a hyperbola.<\/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<h5 class=\"wp-block-heading has-vivid-purple-color has-text-color has-link-color wp-elements-d448cfcb155e36b2660116e91884cabc\"><strong>General equation of a conic&nbsp; <\/strong><\/h5>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Let\\ the\\ focus\\ be\\ S(x_1,\\ y_1)\\ and\\ directrix\\ be\\ the\\ line\\ a\\ x\\ +\\ b\\ y\\ +\\ c\\ =\\ 0\\ \\hspace{10cm}\\]<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\">\\[P(x,\\ y)\\ be\\ any\\ point\\ on\\ it\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[SP\\ =\\ \\sqrt{(x\\ -\\ x_1)^2\\ +\\ (y\\ -\\ y_1)^2}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[PM\\   =\\  Perpendicular\\ distance\\ of\\ P\\ from\\ the\\ line\\  a\\ x\\ +\\ b\\ y\\ +\\ c\\ =\\ 0\\ \\hspace{10cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\pm\\ \\frac{a\\ x\\ +\\ b\\ y\\ +\\ c}{\\sqrt{a^2\\ +\\ b^2}}\\ \\hspace{15cm}\\]  <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Always\\ \\frac{SP}{PM}\\ =\\ e\\ \\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\">\\[\\text{Squaring on both sides}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[[\\sqrt{(x\\ -\\ x_1)^2\\ +\\ (y\\ -\\ y_1)^2}]^2\\ =\\ e^2\\ [\\frac{(ax\\ +\\ by\\ +\\ c)^2}{a^2\\ +\\ b^2}]\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{On simplification we get an equation of the second degree of the form}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ax^2\\ +\\ 2hxy\\ +\\ by^2\\ +\\ 2g\\ x\\ +\\ 2f\\ y +\\ c\\ =\\ 0\\]\n<\/div>\n\n\n\n<h6 class=\"wp-block-heading has-vivid-purple-color has-text-color has-link-color wp-elements-b454d84667953e1709544a1cafcaf5d2\">Note:<\/h6>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{General equation of a conic}\\ ax^2\\ +\\ 2hxy\\ +\\ by^2\\ +\\ 2g\\ x\\ +\\ 2f\\ y +\\ c\\ =\\ 0\\ represents\\]\n<\/div>\n\n\n\n<p>(i) a circle if a = b and h = 0.<\/p>\n\n\n\n<p>(ii) a parabola if &nbsp;h<sup>2 <\/sup>= ab.<\/p>\n\n\n\n<p>(iii) an ellipse if    h<sup>2 <\/sup>&lt; ab.<\/p>\n\n\n\n<p>(iv) a hyperbola if    h<sup>2<\/sup>&gt; ab.<\/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\">\\[\\color {purple} {Example\\ 1\\ .}\\ \\color {red} {Prove\\ that}\\ the\\ equation\\  x^2\\ +\\ 6\\ x\\ y\\ +\\ 9y^2\\ +\\ 4\\ x\\ +\\ 12\\ y\\ -\\ 5\\ =\\ 0\\ \\hspace{10cm}\\]\\[\\color {red} {is\\ a\\ parobala}\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ x^2\\ +\\ 6\\ x\\ y\\ +\\ 9y^2\\ +\\ 4\\ x\\ +\\ 12\\ y\\ -\\ 5\\ =\\ 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\\ =\\ 9\\  \\hspace{5cm}\\ 2h\\ =\\ 6,\\ \\implies\\ h\\ =\\ 3\\  \\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\">\\[\\therefore\\ (1)\\ represents\\ a\\ parabola\\ \\hspace{10cm}\\]<\/div>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/-_VhVPnPerY\" title=\"Conics - Part - 1\" 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<!-- 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\">\\[\\color {purple} {Example\\ 2\\ .}\\ \\color {red} {Show\\ that}\\ the\\ equation\\  x^2\\ +\\ 4y^2\\ -\\ 4\\ x\\ +\\ 24\\ y\\ +\\ 31\\ =\\ 0\\ \\hspace{10cm}\\]\\[\\color {red} {represents\\ an\\ ellipse}\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ x^2\\ +\\ 4y^2\\ -\\ 4\\ x\\ +\\ 24\\ y\\ +\\ 31\\ =\\ 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\\ =\\ 1,\\ b\\ =\\ 4\\  \\hspace{5cm}\\ 2h\\ =\\ 0,\\ \\implies\\ h\\ =\\ 0\\  \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ h^2\\ -\\ ab\\ =\\ (0)^2\\ -\\ 1(4)\\  =\\ -\\ 4\\ \\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=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/cym41qpKlGw\" title=\"CONICS - 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=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 3\\ .}\\ \\color {red} {Check}\\ whether\\ the\\ conic\\ 2\\ x^2\\ -\\ 16\\ x\\ y\\ +\\   8\\ y^2\\ -\\ y\\ +\\ 3\\ = 0\\ \\color {red} {represent\\ a\\ hyperbola}\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ 2\\ x^2\\ -\\ 16\\ x\\ y\\ +\\   8\\ y^2\\ -\\ y\\ +\\ 3\\ &#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\\ =\\ 2,\\ b\\ =\\ 8\\  \\hspace{5cm}\\ 2h\\ =\\ &#8211; 16,\\ \\implies\\ h\\ =\\ -\\ 8\\  \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ h^2\\ -\\ ab\\ =\\ (-8)^2\\ -\\ 2(8)\\ =\\ 64\\ -\\ 16\\ =\\ 48\\ \\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=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/rTVROmkF_kM\" title=\"CONICS - 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=\"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=\"aicp\"><script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js\"><\/script>\n<ins class=\"adsbygoogle\" style=\"display:block\" data-ad-format=\"fluid\" data-ad-layout-key=\"-6t+ed+2i-1n-4w\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"9770958327\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 4\\ .}\\ \\color {red} {Find\\ the\\ equation\\ of\\ the\\ parabola}\\ with\\ focus\\ at\\ (1,\\ -1)\\ \\hspace{10cm}\\]\\[and\\ directrix\\ x\\ -\\ y\\ =\\ 0.\\ \\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,\\ -\\ 1)\\ and\\ directrix\\ is\\ x\\ -\\ 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\\ +\\ 1)^2}}{\\pm\\ \\frac{x\\ -\\ y}{\\sqrt{(1)^2\\ +\\ (-1)^2}}}\\ =\\ 1\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\sqrt{(x\\ -\\ 1)^2\\ +\\ (y\\ +\\ 1)^2}\\ =\\ \\pm\\ \\frac{x\\ -\\ y}{\\sqrt{2}}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(x\\ -\\ 1)^2\\ +\\ (y\\ +\\ 1)^2\\ =\\ \\frac{(x\\ -\\ y)^2}{2}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ 2(x^2\\ -\\ 2\\ x\\ +\\ 1\\ +\\ y^2\\ +\\ 2\\ y\\ +\\ 1)\\ =\\ x^2\\ +\\ y^2\\ -\\ 2\\ x\\ y\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ 2\\ x^2\\ -\\ 4\\ x\\ +\\ 2\\ +\\ 2\\ y^2\\ +\\ 4\\ y\\ +\\ 2\\ -\\ x^2\\ -\\ y^2\\ +\\ 2\\ x\\ y\\ =\\ 0\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ 2\\ x^2\\ -\\ 4\\ x\\ +\\ 2\\ +\\ 2\\ y^2\\ +\\ 4\\ y\\ +\\ 2\\ -\\ x^2\\ -\\ y^2\\ +\\ 2\\ x\\ y\\ =\\ 0\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ \\boxed {x^2\\ +\\ 2\\ x\\ y\\ -\\ 4\\ x\\ +\\ y^2\\ +\\ 4\\ y\\ +\\ 4\\ =\\ 0}\\ \\hspace{8cm}\\]<\/div>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/cIi5PqPbBJc\" title=\"CONICS - PART - 5\" 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} {Example\\ 5\\ .}\\ \\color {red} {Find\\ the\\ equation\\ of\\ the\\ parabola}\\ with\\ focus\\ at\\ (2,\\ 1)\\ \\hspace{10cm}\\]\\[and\\ directrix\\ 2x\\ +\\ y\\ +\\ 1\\ =\\ 0.\\ \\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(2,\\ 1)\\ and\\ directrix\\ is\\ 2\\ x\\ +\\ y\\ +\\ 1\\ =\\ 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\\ -\\ 2)^2\\ +\\ (y\\ -\\ 1)^2}}{\\pm\\ \\frac{2x\\ +\\ y\\ +\\ 1}{\\sqrt{(2)^2\\ +\\ (1)^2}}}\\ =\\ 1\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\sqrt{(x\\ -\\ 2)^2\\ +\\ (y\\ -\\ 1)^2}\\ =\\ \\pm\\ \\frac{2x\\ +\\ y\\ +\\ 1}{\\sqrt{5}}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(x\\ -\\ 2)^2\\ +\\ (y\\ -\\ 1)^2\\ =\\ \\frac{(2x\\ +\\ y\\ +\\ 1)^2}{5}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ 5(x^2\\ -\\ 4\\ x\\ +\\ 4\\ +\\ y^2\\ -\\ 2\\ y\\ +\\ 1)\\ =\\ 4x^2\\ +\\ y^2\\ +\\  1\\ +\\ 4\\ x\\ y\\ + 4\\ x\\ +\\ 2y\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ 5\\ x^2\\ -\\ 20\\ x\\ +\\ 20\\ +\\ 5\\ y^2\\ -\\ 10\\ y\\ +\\ 5\\ -\\ 4\\ x^2\\ -\\ y^2\\ -\\ 1\\ -\\ 4\\ x\\ y\\ -\\ 4x\\ -\\ 2y\\ =\\ 0\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ x^2\\ +\\  4\\ y^2\\ -\\ 4\\ x\\ y\\ -\\ 24\\ x\\   -\\ 12\\ y\\ +\\ 24\\ =\\ 0\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ \\boxed {x^2\\ +\\  4\\ y^2\\ -\\ 4\\ x\\ y\\ -\\ 24\\ x\\   -\\ 12\\ y\\ +\\ 24\\ =\\ 0}\\ \\hspace{8cm}\\]<\/div>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/QFvbqmEJSPg\" title=\"Conics - Part - 6\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<h5 class=\"wp-block-heading has-vivid-purple-color has-text-color has-link-color wp-elements-a00cde5a14ce007fe809efa4c74a9ee4\"><strong>Standard equation of a parabola&nbsp;with vertex at the origin<\/strong><\/h5>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{Standard equation of a parabola with vertex at the origin is}\\ y^2\\ =\\ 4a\\ x\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<figure class=\"wp-block-image size-large\"><img data-recalc-dims=\"1\" fetchpriority=\"high\" decoding=\"async\" width=\"840\" height=\"787\" data-attachment-id=\"53800\" data-permalink=\"https:\/\/yanamtakshashila.com\/?attachment_id=53800\" data-orig-file=\"https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2023\/12\/IMG_0695.jpg?fit=2767%2C2591&amp;ssl=1\" data-orig-size=\"2767,2591\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;1.6&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;iPhone 13&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;1734013113&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;5.1&quot;,&quot;iso&quot;:&quot;100&quot;,&quot;shutter_speed&quot;:&quot;0.02&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;1&quot;}\" data-image-title=\"IMG_0695\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2023\/12\/IMG_0695.jpg?fit=840%2C787&amp;ssl=1\" src=\"https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2023\/12\/IMG_0695.jpg?resize=840%2C787&#038;ssl=1\" alt=\"\" class=\"wp-image-53800\" srcset=\"https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2023\/12\/IMG_0695.jpg?resize=840%2C787&amp;ssl=1 840w, https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2023\/12\/IMG_0695.jpg?resize=390%2C365&amp;ssl=1 390w, https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2023\/12\/IMG_0695.jpg?resize=768%2C719&amp;ssl=1 768w, https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2023\/12\/IMG_0695.jpg?resize=1536%2C1438&amp;ssl=1 1536w, https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2023\/12\/IMG_0695.jpg?resize=2048%2C1918&amp;ssl=1 2048w, https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2023\/12\/IMG_0695.jpg?resize=1200%2C1124&amp;ssl=1 1200w\" sizes=\"(max-width: 840px) 100vw, 840px\" \/><\/figure>\n\n\n\n<p><strong>Focus:<\/strong> The fixed point used to draw the parabola is called the focus (F). Here focus is F(a, 0)<\/p>\n\n\n\n<p style=\"padding-top:0;padding-right:var(--wp--preset--spacing--20);padding-bottom:0;padding-left:var(--wp--preset--spacing--20)\"><strong>Directrix<\/strong>: The fixed line used to draw a parabola is called the directrix of the parabola. Here the equation of the            directrix is x = &#8211; a<\/p>\n\n\n\n<p><strong>Axis<\/strong>: The axis of the parabola is the axis of symmetry.  The curve <mathml>\\[y^2 = 4ax\\] <\/mathml>is            symmetrical about x &#8211; axis and hence x-axis or y = 0 is the axis of the parabola <mathml>\\[y^2 = 4ax\\]<\/mathml>.  Note that the axis of the parabola passes through the focus and perpendicular to the directrix. <\/p>\n\n\n\n<p><strong>Vertex<\/strong>:  The point of intersection of the parabola with its axis is called its vertex.  Here the vertex V is (0, 0)<\/p>\n\n\n\n<p><strong>Focal distance:<\/strong>  The distance between a point on the parabola and its focus is called a focal distance.<\/p>\n\n\n\n<p><strong>Focal chord:<\/strong>  A chord which passes through the focus of the parabola is called the focal chord of the parabola.<\/p>\n\n\n\n<p><strong>Latus rectum<\/strong>:  It is a focal chord perpendicular to the axis of the parabola.  Here, the equation of the lotus rectum is x = a.   Length of the lotus rectum is 4a.<\/p>\n\n\n\n<p>(i).  Vertex O (0, 0)<\/p>\n\n\n\n<p>(ii). Focus S (a, 0)<\/p>\n\n\n\n<p>(iii). Equation of axis is equation x &#8211; axis is y = 0.<\/p>\n\n\n\n<p>(iv). LL&#8217; &#8211; Length of lotus rectum = 4a<\/p>\n\n\n\n<p>(v). Equation of lotus rectum. x = a<\/p>\n\n\n\n<p>(vi). AB &#8211; Directrix<\/p>\n\n\n\n<p>(vii). Equation of the directrix x = &#8211; a.<\/p>\n\n\n\n<h5 class=\"wp-block-heading has-vivid-purple-color has-text-color has-link-color wp-elements-44c791affc32fceb83c1b1de0d43bf7a\"><strong>Standard equation of a parabola&nbsp;with vertex at (h , k) <\/strong><\/h5>\n\n\n\n<figure class=\"wp-block-image size-large\"><img data-recalc-dims=\"1\" decoding=\"async\" width=\"840\" height=\"605\" data-attachment-id=\"53828\" data-permalink=\"https:\/\/yanamtakshashila.com\/?attachment_id=53828\" data-orig-file=\"https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2023\/12\/IMG_0696.jpg?fit=2869%2C2067&amp;ssl=1\" data-orig-size=\"2869,2067\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;1.6&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;iPhone 13&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;1734085734&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;5.1&quot;,&quot;iso&quot;:&quot;100&quot;,&quot;shutter_speed&quot;:&quot;0.02&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;1&quot;}\" data-image-title=\"IMG_0696\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2023\/12\/IMG_0696.jpg?fit=840%2C605&amp;ssl=1\" src=\"https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2023\/12\/IMG_0696.jpg?resize=840%2C605&#038;ssl=1\" alt=\"\" class=\"wp-image-53828\" srcset=\"https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2023\/12\/IMG_0696.jpg?resize=840%2C605&amp;ssl=1 840w, https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2023\/12\/IMG_0696.jpg?resize=507%2C365&amp;ssl=1 507w, https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2023\/12\/IMG_0696.jpg?resize=768%2C553&amp;ssl=1 768w, https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2023\/12\/IMG_0696.jpg?resize=1536%2C1107&amp;ssl=1 1536w, https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2023\/12\/IMG_0696.jpg?resize=2048%2C1476&amp;ssl=1 2048w, https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2023\/12\/IMG_0696.jpg?resize=1200%2C865&amp;ssl=1 1200w\" sizes=\"(max-width: 840px) 100vw, 840px\" \/><\/figure>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{Standard equation of a parabola with vertex at (h, k) is}\\ (y\\ -\\ k)^2\\ =\\ 4a (x &#8211; h)\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 6\\ .}\\ \\color {red} {Find\\ the\\ equation\\ of\\ the\\ Ellipse}\\ with\\ focus\\ (2,\\ 3)\\ and\\ directrix\\ x\\ =\\ 7\\ and\\ e\\ =\\ \\frac{1}{2}\\ \\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(2,\\ 3)\\ and\\ directrix\\ is\\ x\\ -\\ 7\\ =\\ 0,\\ e\\ =\\ \\frac{1}{2}\\ \\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\\ -\\ 2)^2\\ +\\ (y\\ -\\ 3)^2}}{\\pm\\ \\frac{x\\ -\\ 7}{\\sqrt{(1)^2\\ +\\ (0)^2}}}\\ =\\ \\frac{1}{2}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2\\ \\sqrt{(x\\ -\\ 2)^2\\ +\\ (y\\ -\\ 3)^2}\\ =\\ \\pm\\ \\frac{x\\ -\\ 7}{\\sqrt{1}}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[4\\ (x\\ -\\ 2)^2\\ +\\ (y\\ -\\ 3)^2\\ =\\ \\frac{(x\\ -\\ 7)^2}{1}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ 4(x^2\\ -\\ 4\\ x\\ +\\ 4\\ +\\ y^2\\ -\\ 6\\ y\\ +\\ 9)\\ =\\ x^2\\ -\\ 14\\ x\\ +\\ 49\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ 4\\ x^2\\ -\\ 16\\ x\\ +\\ 16\\ -\\ 24\\ y\\ +\\ 36\\ -\\ x^2\\ +\\ 14\\ x\\ -\\ 49\\ =\\ 0\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ 3\\ x^2\\ -\\ 2\\ x\\  +\\  4\\ y^2\\ -\\ 24\\ y\\ +\\ 3\\ =\\ 0\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ 3\\ x^2\\ +\\  4\\ y^2\\ &#8211; 2\\ x\\ -\\ 24\\ y\\ +\\ 3\\ =\\ 0\\ \\hspace{8cm}\\]<\/div>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/zUyFA-arQRw\" title=\"CONICS - 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=\"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<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Syllabus: General equation of conics&nbsp;\u2013&nbsp;Classification of conics&nbsp;\u2013&nbsp;Standard equations of parabola&nbsp;\u2013&nbsp;Vertex, focus, axis, directrix, focal distance, focal chord, latus-rectum of parabola&nbsp;, Standard equations of ellipse&nbsp;\u2013&nbsp;Vertices, foci, major axis, minor axis, directrices, eccentricity, centre and latus-rectums of ellipse&nbsp;\u2013&nbsp;Simple problems. Conic: A conic is defined as the locus of a point which moves such that its distance from [&hellip;]<\/p>\n","protected":false},"author":187055548,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"set","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2},"_wpas_customize_per_network":false,"jetpack_post_was_ever_published":false},"categories":[711788742,711788741],"tags":[],"class_list":["post-44805","post","type-post","status-publish","format-standard","hentry","category-applied-mathematics-ii","category-new-regulation-2023"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>COORDINATE GEOMETRY (UNIT - 1 FOR CIRCUIT AND UNIT - 2 FOR NON - CIRCUIT) - YANAMTAKSHASHILA<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/yanamtakshashila.com\/?p=44805\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"COORDINATE GEOMETRY (UNIT - 1 FOR CIRCUIT AND UNIT - 2 FOR NON - CIRCUIT) - YANAMTAKSHASHILA\" \/>\n<meta property=\"og:description\" content=\"Syllabus: General equation of conics&nbsp;\u2013&nbsp;Classification of conics&nbsp;\u2013&nbsp;Standard equations of parabola&nbsp;\u2013&nbsp;Vertex, focus, axis, directrix, focal distance, focal chord, latus-rectum of parabola&nbsp;, Standard equations of ellipse&nbsp;\u2013&nbsp;Vertices, foci, major axis, minor axis, directrices, eccentricity, centre and latus-rectums of ellipse&nbsp;\u2013&nbsp;Simple problems. 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