{"id":44893,"date":"2023-12-23T13:23:41","date_gmt":"2023-12-23T07:53:41","guid":{"rendered":"https:\/\/viswaguide.com\/?p=44893"},"modified":"2025-12-25T21:20:56","modified_gmt":"2025-12-25T15:50:56","slug":"coordinate-geometry-non-circuit","status":"publish","type":"post","link":"https:\/\/viswaguide.com\/?p=44893","title":{"rendered":"COORDINATE GEOMETRY &#8211; 1 (Non Circuit Branches)"},"content":{"rendered":"\n<h5 class=\"wp-block-heading has-vivid-purple-color has-text-color has-link-color wp-elements-bb1f69e4a56dc18d1126026da62c13dd\">SYLLABUS<\/h5>\n\n\n\n<p class=\"wp-block-paragraph\">Equation of a circle with given centre and radius &#8211; General equation of circles &#8211; Centre and radius of a circle from general equation &#8211; Concentric circles &#8211; Contact of circles &#8211; Orthogonal circles &#8211; Simple problems.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Definition:<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">A circle is the locus of a point which moves in a plane in such a way that its distance from a fixed point remains constant.&nbsp;&nbsp; The fixed point is called the <strong>centre of the circle<\/strong> and the constant distance is called the <strong>radius<\/strong> of the circle.<\/p>\n\n\n\n<figure class=\"wp-block-image aligncenter size-full\"><img data-recalc-dims=\"1\" fetchpriority=\"high\" decoding=\"async\" width=\"589\" height=\"465\" data-attachment-id=\"44898\" data-permalink=\"https:\/\/viswaguide.com\/?attachment_id=44898\" data-orig-file=\"https:\/\/i0.wp.com\/viswaguide.com\/wp-content\/uploads\/2023\/12\/image-3.png?fit=589%2C465&amp;ssl=1\" data-orig-size=\"589,465\" 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=\"image-3\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/i0.wp.com\/viswaguide.com\/wp-content\/uploads\/2023\/12\/image-3.png?fit=589%2C465&amp;ssl=1\" src=\"https:\/\/i0.wp.com\/viswaguide.com\/wp-content\/uploads\/2023\/12\/image-3.png?resize=589%2C465&#038;ssl=1\" alt=\"\" class=\"wp-image-44898\" srcset=\"https:\/\/i0.wp.com\/viswaguide.com\/wp-content\/uploads\/2023\/12\/image-3.png?w=589&amp;ssl=1 589w, https:\/\/i0.wp.com\/viswaguide.com\/wp-content\/uploads\/2023\/12\/image-3.png?resize=462%2C365&amp;ssl=1 462w\" sizes=\"(max-width: 589px) 100vw, 589px\" \/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Equation of the circle with centre (h, k)&nbsp; and radius \u2018r\u2019 units.<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">CP = r&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\sqrt{(x\\ -\\ h)^2\\ +\\ (y\\ -\\ k)^2}\\ =\\ r\\  (Using\\ distance\\ formula)\\]<script id=\"wp-hooks-js\" src=\"https:\/\/viswaguide.com\/wp-includes\/js\/dist\/hooks.min.js?ver=7496969728ca0f95732d\"><\/script>\n<script id=\"wp-i18n-js\" src=\"https:\/\/viswaguide.com\/wp-includes\/js\/dist\/i18n.min.js?ver=781d11515ad3d91786ec\"><\/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 id=\"mathjax-js\" src=\"https:\/\/cdnjs.cloudflare.com\/ajax\/libs\/mathjax\/2.7.7\/MathJax.js?config=TeX-MML-AM_CHTML\"><\/script>\n<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(x\\ -\\ h)^2\\ +\\ (y\\ -\\ k)^2\\ =\\ r^2\\]<\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Note:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The equation of the circle with centre (0, 0 ) and radius \u2018r\u2019 units is&nbsp; x<sup>2&nbsp; <\/sup>+ y<sup>2<\/sup> &nbsp;= r<sup>2<\/sup><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 1\\ .}\\ \\color {red} {What\\ is\\ the\\ equation\\ of\\ the\\ circle}\\ with\\ centre\\ at\\ origin\\ and\\ radius\\ 1\\ unit?\\ \\hspace{15cm}\\]<\/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\">\\[We\\ know\\ that\\ the\\ equation\\ of\\ circle\\ is\\ (x\\ -\\ h)^2\\ +\\ (y\\ -\\ k)^2\\ =\\ r^2\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Given\\ h\\ =\\ 0,\\ k = 0\\ and\\ r\\ =\\ 1\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(x\\ -\\ 0)^2\\ +\\ (y\\ -\\ 0)^2\\ =\\ 1^2\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[x^2\\ +\\ y^2\\ =\\ 1\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[The\\ equation\\ of\\ the\\ circle\\ is\\ \\boxed{x^2\\ +\\ y^2\\ -\\ 1=\\  0}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/URkjABHomZQ\" title=\"Analytical Geometry - 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=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 2\\ .}\\ \\color {red} {Find\\ the\\ equation\\ of\\ the\\ circle}\\ with\\ centre\\ (-5, 7)\\ and\\ radius\\  5\\ units.\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} { Soln:}\\ Given\\ centre\\ =\\ (-5, 7)\\ \\hspace{4cm}\\ and\\ radius\\ =\\ 5\\ \\hspace{6cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[We\\ know\\ that\\ the\\ equation\\ of\\ circle\\ is\\ (x\\ -\\ h)^2\\ +\\ (y\\ -\\ k)^2\\ =\\ r^2\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Given\\ h\\ =\\ -\\ 5,\\ k = 7\\ and\\ r\\ =\\ 5\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(x\\ +\\ 5)^2\\ +\\ (y\\ -\\ 7)^2\\ =\\ 5^2\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[x^2\\ +\\ 10\\ x\\ +\\ 25\\ +\\ y^2\\ -\\ 14y\\ +\\ 49\\ =\\ 25\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[x^2\\ +\\  y^2\\  +\\ 10\\ x\\ -\\  14y\\ +\\ 25\\ +\\ 49\\ -\\ 25\\ =\\ 0\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[x^2\\ +\\  y^2\\  +\\ 10\\ x\\ -\\  14y\\ +\\ 49\\ =\\ 0\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\therefore\\ \nThe\\ required\\ equation\\ of\\ the\\ circle\\ is\\ \\hspace{7cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{x^2\\ +\\  y^2\\  +\\   10\\  x\\ -\\ 14\\ y\\ +\\ 49\\ =\\ 0}\\ \\hspace{5cm}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/itEyeO4CO0w\" title=\"Analytical Geometry - 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=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 3\\ .}\\ \\color {red} {Write\\ down\\ the\\ equation\\ of\\ the\\ circle}\\ whose\\ centre\\ is\\ (0, &#8211; 2)\\ and\\ radius\\  5.\\\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} { Soln:}\\ Given\\ centre\\ =\\ (0, -2)\\ \\hspace{4cm}\\ and\\ radius\\ =\\ 5\\ \\hspace{6cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[We\\ know\\ that\\ the\\ equation\\ of\\ circle\\ is\\ (x\\ -\\ h)^2\\ +\\ (y\\ -\\ k)^2\\ =\\ r^2\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Given\\ h\\ =\\ 0,\\ k = -\\ 2\\ and\\ r\\ =\\ 5\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(x\\ -\\ 0)^2\\ +\\ (y\\ +\\ 2)^2\\ =\\ 5^2\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[x^2\\ +\\ y^2\\ +\\ 4y\\ +\\ 4\\ =\\ 25\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[x^2\\ +\\  y^2\\  +\\ 4y\\ +\\ 4\\ -\\ 25\\ =\\ 0\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[x^2\\ +\\  y^2\\  +\\ 4y\\ -\\ 21\\ =\\ 0\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\therefore\\ \nThe\\ required\\ equation\\ of\\ the\\ circle\\ is\\ \\hspace{7cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{x^2\\ +\\  y^2\\  +\\ 4y\\ -\\ 21\\ =\\ 0}\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>General equation of the circle:&nbsp;&nbsp;&nbsp;&nbsp; <\/strong>x<sup>2<\/sup>&nbsp; + &nbsp;&nbsp;y<sup>2 <\/sup>&nbsp;+ 2gx&nbsp; + 2fy&nbsp; + c = 0<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Centre = (-g , -f )&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; and&nbsp;&nbsp;&nbsp; radius&nbsp;&nbsp;&nbsp;&nbsp; r = &nbsp;\u221a( g<sup>2<\/sup>&nbsp; + f<sup>2<\/sup> \u2013 c)<\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 4\\ .}\\ \\color {red} {Find\\ the\\ centre\\ and\\ radius\\ of\\ the\\ circle}\\ x^2\\ +\\ y^2\\ +\\ 2\\ x\\ +\\ 2\\ y\\ -\\ 7\\ =\\ 0\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} { Soln:}\\ Given\\ x^2\\ +\\ y^2\\ +\\ 2\\ x\\ +\\ 2\\ y\\ -\\ 7\\ =\\ 0\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[We\\ know\\ that\\ the\\ equation\\ of\\ circle\\ is\\ x^2\\ +\\ y^2\\ +\\ 2\\ g\\ x\\ +\\ 2\\ f\\ y\\ +\\ c\\ =\\ 0\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2\\ g\\ =\\ 2\\ \\hspace{3cm}\\ 2\\ f\\ =\\ 2\\ \\hspace{3cm}\\ c\\ =\\ -7\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[g\\ =\\ 1\\ \\hspace{3cm}\\ f\\ =\\ 1\\ \\hspace{3cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[centre\\ =\\ (-\\ g,\\  -\\ f)\\ \\hspace{4cm}\\ r\\ =\\ \\sqrt{(g^2\\ +\\ f^2\\ -\\ c)}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[centre\\ =\\ (-\\ 1,\\  -\\ 1)\\ \\hspace{4cm}\\ r\\ =\\ \\sqrt{(1^2\\ +\\ 1^2\\ +\\ 7)}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{6cm}\\ r\\ =\\ \\sqrt{(1\\ +\\ 1\\ +\\ 7)}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{6cm}\\ r\\ =\\ \\sqrt{9}\\ =\\ 3\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\fbox{centre  =    (- 1,  &#8211; 1)                           r  = 3}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/JQ0JWPz59U8\" title=\"Analytical Geometry - Part - 3\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 5\\ .}\\ \\color {red} {Find\\ the\\ centre\\ and\\ radius\\ of\\ the\\ circle}\\ x^2\\ +\\ y^2\\  -\\ 8\\ y\\ +\\ 3\\ =\\ 0\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ Given\\ x^2\\ +\\ y^2\\ -\\ 8\\ y\\ +\\ 3\\ =\\ 0\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[We\\ know\\ that\\ the\\ equation\\ of\\ circle\\ is\\ x^2\\ +\\ y^2\\ +\\ 2\\ g\\ x\\ +\\ 2\\ f\\ y\\ +\\ c\\ =\\ 0\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2\\ g\\ =\\ 0\\ \\hspace{3cm}\\ 2\\ f\\ =\\ -\\ 8\\ \\hspace{3cm}\\ c\\ =\\ 3\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[g\\ =\\ 0\\ \\hspace{3cm}\\ f\\ =\\ -\\ 4\\ \\hspace{3cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[centre\\ =\\ (-\\ g,\\  -\\ f)\\ \\hspace{4cm}\\ r\\ =\\ \\sqrt{(g^2\\ +\\ f^2\\ -\\ c)}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[centre\\ =\\ (0,\\  4)\\ \\hspace{4cm}\\ r\\ =\\ \\sqrt{(0^2\\ +\\ (-4)^2\\ -\\ 3)}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{6cm}\\ r\\ =\\ \\sqrt{(0\\ +\\ 16\\ -\\ 3)}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{6cm}\\ r\\ =\\ \\sqrt{13}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[centre  =    (0,  4)\\ \\hspace{5cm}\\                           r\\ =\\ \\sqrt{13}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/nBvqNJIAneQ\" title=\"Analytical Geometry - Part - 5\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><strong><u>Contact of Circles:<\/u><\/strong><\/h3>\n\n\n\n<figure class=\"wp-block-image aligncenter size-full\"><img data-recalc-dims=\"1\" decoding=\"async\" width=\"204\" height=\"128\" data-attachment-id=\"44968\" data-permalink=\"https:\/\/viswaguide.com\/?attachment_id=44968\" data-orig-file=\"https:\/\/i0.wp.com\/viswaguide.com\/wp-content\/uploads\/2023\/12\/image-5.png?fit=204%2C128&amp;ssl=1\" data-orig-size=\"204,128\" 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=\"image-5\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/i0.wp.com\/viswaguide.com\/wp-content\/uploads\/2023\/12\/image-5.png?fit=204%2C128&amp;ssl=1\" src=\"https:\/\/i0.wp.com\/viswaguide.com\/wp-content\/uploads\/2023\/12\/image-5.png?resize=204%2C128&#038;ssl=1\" alt=\"\" class=\"wp-image-44968\"\/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">Case ( i ) :<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Two circles touch externally if the distance between their centres is equal to sum of their radii.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">i.e &nbsp;<strong>C<sub>1<\/sub>C<sub>2&nbsp; <\/sub>=&nbsp; r<sub>1<\/sub>&nbsp; +&nbsp; r<sub>2<\/sub><\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Case ( ii ) :<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Two circles touch internally if the distance between their centres is equal to difference of their radii.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">i.e&nbsp; <strong>C<sub>1<\/sub>C<sub>2&nbsp; <\/sub>= r<sub>1 <\/sub>&nbsp;&#8211;&nbsp; r<sub>2&nbsp; <\/sub><\/strong>or&nbsp;&nbsp; <strong>r<sub>2<\/sub>&nbsp; &#8211;&nbsp; <strong>r<sub>1<\/sub><\/strong><\/strong><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 6\\ .}\\ \\color {red} {Prove\\ that}\\ the\\ circles\\ x^2\\ +\\ y^2\\ -\\ 4x\\ -\\ 6y\\ +\\ 9\\ = 0\\ \\hspace{5cm}\\]\\[ and\\ x^2\\ +\\  y^2\\ +\\ 2x\\ +\\ 2y\\ -\\ 7\\ = 0\\ touch\\ each\\ other.\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ \\hspace {19cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Given\\ x^2 + y^2 -\\ 4x\\ -\\ 6y\\ +\\ 9\\ = 0 &#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212; (1)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Given\\ x^2\\ +\\  y^2\\ +\\ 2x\\ +\\ 2y\\ -\\ 7\\ = 0 &#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212; (2)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[From\\ (1)\\ \\hspace 10cm\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2g_1 =\\ -\\ 4\\ \\hspace 2cm\\  2f_1\\ =\\ -\\ 6\\ \\hspace 2cm\\ c_1 =\\ 9\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[g_1 =\\ -\\ 2\\ \\hspace 2cm\\  f_1 =\\ -\\ 3\\ \\hspace 2cm\\ c_1 =\\ 9\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Centre\\ is\\  C_1 = (-g_1,\\ -f_1)\\ \\hspace 10cm\\ r_1 = \\sqrt{g_1^2 + f_1^2 -c_1}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ C_1 = (2,\\ \\ 3)\\ \\hspace 10cm\\ r_1 = \\sqrt{(-2)^2\\ +\\ (-3)^2\\  -\\  9}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\hspace 10cm\\ r_1 = \\sqrt{4\\ +\\ 9\\  -\\ 9}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\hspace 10cm\\ r_1 = \\sqrt{4}\\ =\\  2\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{C_1 = ( 2, 3)\\  and\\ r_1 =\\ 2}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[From\\ (2)\\ \\hspace 10cm\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2g_2 =\\ 2\\ \\hspace 2cm\\  2f_2\\ =\\ 2\\ \\hspace 2cm\\ c_2 =\\ -\\ 7\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[g_2 =\\ 1\\ \\hspace 2cm\\  f_2 =\\ 1\\ \\hspace 2cm\\ c_2 =\\  -\\ 7\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Centre\\ is\\  C_2 = (-g_2,\\ -f_2)\\ \\hspace 10cm\\ r_2 = \\sqrt{g_2^2 + f_2^2 -c_2}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ C_2 = (-\\ 1\\ , -\\ 1)\\ \\hspace 10cm\\ r_2 = \\sqrt{(1)^2\\ +\\ (1)^2\\  +\\ 7}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\hspace 10cm\\ r_2\\ = \\sqrt{1\\ + 1\\  +\\ 7}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\hspace 10cm\\ r_2 = \\sqrt{9} =\\ 3\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{C_2 = (-\\ 1, -\\ 1)\\  and\\ r_2 =\\ 3}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[C_1C_2 = \\sqrt{(-1\\ -\\ 2)^2\\ +\\ (-1\\ -\\  3)^2}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[C_1C_2 = \\sqrt{(-3)^2\\ +\\ (-4)^2}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[C_1C_2 = \\sqrt{9\\ +\\ 16}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[C_1C_2 = \\sqrt{25}\\ =\\ 5\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[r_1\\ +\\  r_2\\ =\\ 2\\ +\\ 3\\ =\\ 5\\ =\\  C_1C_2\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{C_1C_2 = r_1 + r_2}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[The\\ given\\ circles\\ touch\\ each\\ other\\ externally\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 7\\ .}\\ \\color {red} {Prove\\ that\\ the\\ circles}\\ x^2\\ +\\ y^2\\ -\\ 10\\ x\\ -\\ 24\\ y\\ +\\ 120\\ =\\ 0\\ and\\ \\hspace{7cm}\\]\\[x^2\\ +\\ y^2\\ =\\ 400\\ \\color {red} {touch\\ each\\ other}\\ \\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\">\\[Given\\ x^2\\ +\\ y^2\\ -\\ 10\\ x\\ -\\ 24\\ y\\ +\\ 120\\ =\\ 0 &#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212; (1)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Given\\ x^2\\ +\\ y^2\\ =\\ 400 &#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212; (2)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[From\\ (1)\\ \\hspace 10cm\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2g_1 = &#8211; 10\\ \\hspace 2cm\\  2f_1 = &#8211; 24\\ \\hspace 2cm\\ c_1 = 120\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[g_1 =\\ &#8211; 5\\ \\hspace 2cm\\  f_1 =\\ &#8211;  12\\ \\hspace 2cm\\ c_1 = 120\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Centre\\ is\\  C_1 = (-g_1,\\ -f_1)\\ \\hspace 10cm\\ r_1 = \\sqrt{g_1^2 + f_1^2 -c_1}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ C_1 = (5,\\ 12)\\ \\hspace 10cm\\ r_1 = \\sqrt{(\\ -\\ 5)^2 + (-\\  12)^2\\  -\\ 120}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\hspace 10cm\\ r_1 = \\sqrt{25\\ +\\ 144\\ -\\ 120}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\hspace 10cm\\ r_1 = \\sqrt{169\\ -\\ 120}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\hspace 10cm\\ r_1 = \\sqrt{49} =\\ 7\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{C_1 = ( 5, 12)\\  and\\ r_1 = 7}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[From\\ (2)\\ \\hspace 10cm\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2g_2 = 0\\ \\hspace 2cm\\  2f_2 = 0\\ \\hspace 2cm\\ c_2 = &#8211; 400\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[g_2 =\\ 0 \\hspace 2cm\\  f_2 = 0\\ \\hspace 2cm\\ c_2 = &#8211; 400\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Centre\\ is\\  C_2 = (-g_2,\\ -f_2)\\ \\hspace 10cm\\ r_2 = \\sqrt{g_2^2 + f_2^2 -c_2}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ C_2 = (0,\\ 0)\\ \\hspace 10cm\\ r_2 = \\sqrt{(0)^2 +\\ (0)^2  + 400}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\hspace 10cm\\ r_2 = \\sqrt{0\\ +\\ 0\\  +\\ 400}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\hspace 10cm\\ r_2\\ =\\ \\sqrt{400}\\ =\\ 20\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{C_2 = ( 0, 0)\\  and\\ r_2\\ =\\ 20}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[C_1C_2 = \\sqrt{(0\\ -\\ 5)^2 + (0\\ -\\ 12)^2}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[C_1C_2 = \\sqrt{(-\\ 5)^2 + (-\\ 12)^2}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[C_1C_2 = \\sqrt{25\\ +\\  144}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[C_1C_2 = \\sqrt{169}\\ =\\  13\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[r_2 &#8211; r_1 \\ =\\  20\\ -\\ 13\\ =\\  7\\ = C_1C_2\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{C_1C_2 = r_2 &#8211; r_1}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[The\\ given\\ circles\\ touch\\ each\\ other\\ internally\\]<\/div>\n\n\n\n<iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/oSH4JhvPKj4\" title=\"Analytical Geometry (Exercise) - Part - 20\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n","protected":false},"excerpt":{"rendered":"<p>SYLLABUS Equation of a circle with given centre and radius &#8211; General equation of circles &#8211; Centre and radius of a circle from general equation &#8211; Concentric circles &#8211; Contact of circles &#8211; Orthogonal circles &#8211; Simple problems. Definition: A circle is the locus of a point which moves in a plane in such a [&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 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