{"id":14590,"date":"2021-05-09T11:54:44","date_gmt":"2021-05-09T06:24:44","guid":{"rendered":"https:\/\/yanamtakshashila.com\/?p=14590"},"modified":"2024-04-13T16:06:19","modified_gmt":"2024-04-13T10:36:19","slug":"1-1-n-analytical-geometry-exercise-problems-with-solutions","status":"publish","type":"post","link":"https:\/\/yanamtakshashila.com\/?p=14590","title":{"rendered":"1.1 &#8211; N &#8211; Analytical Geometry Exercise Problems With Solutions"},"content":{"rendered":"\n<p class=\"has-text-align-center\">  <strong><u>Part \u2013 A<\/u><\/strong>  <\/p>\n\n\n\n<p class=\"has-text-align-left\">1.    &nbsp;Find the perpendicular distance from the point (3, -5) to the straight line 3x-4y-26=0.  <\/p>\n\n\n\n<p>Soln:    W.K.T&nbsp; The length of the perpendicular distance from (x<sub>1<\/sub>,y<sub>1<\/sub>) to the line  ax + by + c = 0&nbsp; is <\/p>\n\n\n\n<p>            \u00b1&nbsp; (ax<sub>1<\/sub> + by<sub>1<\/sub> + c)\/ \u221a(a<sup>2<\/sup> + b<sup>2<\/sup> ) <\/p>\n\n\n\n<p>           Given straight line is &nbsp;&nbsp;&nbsp; 3x-4y-26 =0 <\/p>\n\n\n\n<p>            Given point (x<sub>1<\/sub>,y<sub>1<\/sub>)&nbsp; =&nbsp; (3, -5) <\/p>\n\n\n\n<p>             i.e&nbsp; \u00b1&nbsp;  3(3) &#8211; 4(-5) &#8211; 26\/ \u221a(3<sup>2<\/sup> + (-4)<sup>2<\/sup> )&nbsp;&nbsp;&nbsp;=&nbsp;&nbsp; 3 \/&nbsp;5 <\/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<!-- admitad.banner: 87u3svniba109003a6887d95a12660 Lightinthebox WW -->\n<a target=\"_blank\" rel=\"nofollow noopener\" href=\"https:\/\/ad.admitad.com\/g\/87u3svniba109003a6887d95a12660\/?i=4\"><img fetchpriority=\"high\" decoding=\"async\" width=\"300\" height=\"250\" border=\"0\" src=\"https:\/\/ad.admitad.com\/b\/87u3svniba109003a6887d95a12660\/\" alt=\"Lightinthebox WW\"\/><\/a>\n<!-- \/admitad.banner -->\n\n\n\n<p> 2.&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Find the distance between the line 3x+4y = 9 and 6x+8y  = 15.  <\/p>\n\n\n\n<p>Soln:   W.K.T&nbsp; the&nbsp; distance between the parallel lines <\/p>\n\n\n\n<p>           ax + by + c<sub>1<\/sub> = 0 &nbsp;and&nbsp; ax + by + c<sub>2<\/sub> = 0 &nbsp;is <\/p>\n\n\n\n<p>           \u00b1&nbsp; (c<sub>1&nbsp; <\/sub>&#8211; &nbsp;c<sub>2<\/sub> )\/ &nbsp;\u221a(a<sup>2<\/sup> + b<sup>2<\/sup> ) <\/p>\n\n\n\n<p>             6x+8y  = 15 <\/p>\n\n\n\n<p>             2 (  3x+4y ) = 15<\/p>\n\n\n\n<p>              3x + 4y = 15\/2  <\/p>\n\n\n\n<p>          Here&nbsp; a = 3,&nbsp; b = 4&nbsp; ,&nbsp;&nbsp; c<sub>1<\/sub> =&nbsp; 9&nbsp;&nbsp; and&nbsp;&nbsp; c<sub>2<\/sub> =&nbsp;  15\/2<\/p>\n\n\n\n<p>             i.e&nbsp;&nbsp; 9 &#8211; 15\/2\/\u221a(3<sup>2<\/sup> + 4<sup>2<\/sup> )&nbsp;  &nbsp;&nbsp;&nbsp;=&nbsp;&nbsp; 3 \/10&nbsp;    <\/p>\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<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<p>3.    Show that the lines 3x+2y+9=0 and 12x+8y-15 =0 are parallel.    <\/p>\n\n\n\n<p>Soln:   3x+ 2y+9  = 0 &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(1)   <\/p>\n\n\n\n<p>         12x+2y+14 = 0 &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;        (2)    <\/p>\n\n\n\n<p>         Slope of the line (1) = m<sub>1<\/sub> =&nbsp; &#8211; a\/b <\/p>\n\n\n\n<p>                                                    =&nbsp; &#8211; 3\/2 <\/p>\n\n\n\n<p>          Slope of the line (2) = m<sub>2<\/sub> =&nbsp;  &#8211; a\/b  <\/p>\n\n\n\n<p>                                                     =&nbsp; &#8211; 12\/8<\/p>\n\n\n\n<p>                                                      =&nbsp; &#8211; 3\/2 <\/p>\n\n\n\n<p>                              m<sub>1&nbsp; <\/sub>=&nbsp; m<sub>2<\/sub>     <\/p>\n\n\n\n<p>              \u2234&nbsp; The lines are parallel.&nbsp; <\/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<p>  4.&nbsp;&nbsp;&nbsp; &nbsp;Find \u2018p\u2019 such that the lines 3x+4y = 8 and px + 2y = 7 are parallel.  <\/p>\n\n\n\n<p>  Soln:&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 3x+4y = 8  &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(1) <\/p>\n\n\n\n<p>                           px + 2y = 7  &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;   (2)   <\/p>\n\n\n\n<p>                          Slope of the line (1) = m<sub>1<\/sub> =&nbsp; &#8211; 3\/4 <\/p>\n\n\n\n<p>                          Slope of the line (2) = m<sub>2<\/sub> =&nbsp; &#8211; p\/2<\/p>\n\n\n\n<p>                          Since (1) and (2) are parallel lines <\/p>\n\n\n\n<p>                          m<sub>1&nbsp; <\/sub>=&nbsp; m<sub>2<\/sub> <\/p>\n\n\n\n<p>                        -3\/4&nbsp; =&nbsp;&nbsp;&#8211; p\/2<\/p>\n\n\n\n<p>                            \u2234&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; p&nbsp;&nbsp; =&nbsp;   3\/2<\/p>\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<ins class=\"adsbygoogle\" style=\"display:block\" data-ad-format=\"autorelaxed\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"3040040190\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n<\/div>\n\n\n\n<p>5. &nbsp;&nbsp; Show that the lines 27x-18y+25 =0 and 2x + 3y+7 =0 are perpendicular.   <\/p>\n\n\n\n<p>Soln:&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;  27x-18y+25 =0 &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(1)<\/p>\n\n\n\n<p>                          2x + 3y+7 =0  &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;   (2)   <\/p>\n\n\n\n<p>                          Slope of the line (1) = m<sub>1<\/sub> =&nbsp; &#8211; a\/b <\/p>\n\n\n\n<p>                                                               m<sub>1<\/sub> &nbsp;=&nbsp; &#8211; 27\/-18<\/p>\n\n\n\n<p>                                                               m<sub>1<\/sub> &nbsp;=&nbsp; 3\/2      <\/p>\n\n\n\n<p>                            Slope of the line (2) = m<sub>2<\/sub> =&nbsp; &#8211; 2\/3                               &nbsp;<\/p>\n\n\n\n<p>                           m<sub>1 <\/sub>&nbsp;m<sub>2<\/sub> =&nbsp; ( 3\/2   ) \u00d7&nbsp; ( &#8211; 2\/3  )    <\/p>\n\n\n\n<p>                                           =&nbsp; &#8211; 1 <\/p>\n\n\n\n<p>                             \u2234&nbsp; The lines (1) and (2) are perpendicular. <\/p>\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<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<p> 6.&nbsp;&nbsp;&nbsp;&nbsp;Find the value of k if the lines 2x + ky -11 =0 &nbsp;and &nbsp;5x &#8211; 3y + 4 = 0 &nbsp;are perpendicular.  <\/p>\n\n\n\n<p>Soln:                       2x +  ky -11  = 0 &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;  (1) <\/p>\n\n\n\n<p>                               5x &#8211; 3y + 4 = 0 &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;     (2) <\/p>\n\n\n\n<p>                              Slope of the line (1) = m<sub>1<\/sub> =&nbsp; &#8211; 2\/k   <\/p>\n\n\n\n<p>                              Slope of the line (2) = m<sub>2<\/sub> =&nbsp; &#8211; 5\/-3<\/p>\n\n\n\n<p>                                                                   m<sub>2<\/sub> =&nbsp; 5\/ 3 <\/p>\n\n\n\n<p>                               Since (1) and (2) are &nbsp;perpendicular <\/p>\n\n\n\n<p>                               m<sub>1<\/sub> m<sub>2<\/sub>&nbsp; =&nbsp; &#8211; 1   <\/p>\n\n\n\n<p>                                <sub> <\/sub>(- 2\/k ) ( 5\/3) &nbsp;=&nbsp;&nbsp; -1 <\/p>\n\n\n\n<p>                                ( -10\/3k )&nbsp;&nbsp;&nbsp; =&nbsp; &#8211; 1 <\/p>\n\n\n\n<p>                                    10&nbsp;&nbsp;&nbsp;&nbsp; =&nbsp; 3k <\/p>\n\n\n\n<p>                               \u2234&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; k &nbsp;&nbsp;&nbsp;= 10\/3    <\/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> 7.    Write down the combined equation of the pair of lines  &nbsp;&nbsp;&nbsp; &nbsp;2x + y&nbsp; = 0&nbsp; and&nbsp; 3x &#8211; y = 0. <\/p>\n\n\n\n<p> Soln:&nbsp;&nbsp;&nbsp; The two separate lines are  2x + y &nbsp; = 0&nbsp; and&nbsp;  3x &#8211; y = 0 <\/p>\n\n\n\n<p>               The combined equation is  <\/p>\n\n\n\n<p>                (2x + y) (3x &#8211; y)&nbsp; = 0     <\/p>\n\n\n\n<p>                6x<sup>2<\/sup> &#8211; 2xy + 3xy&nbsp; &#8211; y<sup>2<\/sup> =&nbsp; 0      <\/p>\n\n\n\n<p>6x<sup>2&nbsp; <\/sup>+ xy&nbsp; &#8211; y<sup>2<\/sup> =&nbsp; 0<\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[8.\\ \\color{green}{Find\\ the\\ combined\\ equation\\ of\\ the\\ two\\ straight\\ lines\\ represented\\ by}\\ \\hspace{7cm}\\]\\[\\color{green}{2x\\ +\\ 3y\\ =\\ 0\\ and\\ 4x\\ -\\ 5y\\ =\\ 0}\\ \\hspace{8cm}\\]<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\">\\[\\hspace{15cm}\\ April\\ 2024\\]<\/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{5cm}\\ The\\ two\\ separate\\ lines\\ are\\ 2x\\ +\\ 3y\\ =0\\ and\\ 4x\\ -\\ 5y\\ =\\ 0\\ \\hspace{10cm}\\]\n<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[The\\ combined\\ equation\\ is\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ (2x\\ +\\ y) (4x\\ -\\ 5y)\\&nbsp; =\\ 0\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ 6x^2\\ -\\ 10xy\\ +\\ 4xy\\ -\\ 5y^2\\ =\\ 0\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ 6x^2\\ -\\ 6xy\\ -\\ 5y^2\\ =\\ 0\\]<\/div>\n\n\n\n<p>9. Write down the separate equations of the pair of lines 3y<sup>2<\/sup> + 7xy =&nbsp; 0.<\/p>\n\n\n\n<p>  Soln:&nbsp;&nbsp;  3y<sup>2<\/sup> + 7xy =&nbsp; 0 <\/p>\n\n\n\n<p>                y ( 3y + 7x)  = 0<\/p>\n\n\n\n<p>              y = 0,   3y + 7x = 0    <\/p>\n\n\n\n<p>              \u2234 The separate equations are   y = 0 &nbsp; and&nbsp;&nbsp;&nbsp;  3y + 7x = 0   <\/p>\n\n\n\n<p>10. Find the value of &nbsp;\u2018k\u2019 &nbsp;if the pair of &nbsp;lines kx<sup>2&nbsp; <\/sup>+ 4xy&nbsp; &#8211; 4y<sup>2<\/sup> =&nbsp; 0&nbsp; are perpendicular to each&nbsp; Other.<\/p>\n\n\n\n<p> Soln:&nbsp;&nbsp;&nbsp;  kx<sup>2&nbsp; <\/sup>+ 4xy&nbsp; &#8211; 4y<sup>2<\/sup>  =&nbsp; 0 <\/p>\n\n\n\n<p>              This is of the form&nbsp;&nbsp; ax<sup>2<\/sup>&nbsp; +&nbsp; 2hxy&nbsp; + by<sup>2<\/sup> =&nbsp; 0 <\/p>\n\n\n\n<p>              a = k,&nbsp;&nbsp; 2h = 4&nbsp;&nbsp; &nbsp; h = 2,&nbsp;&nbsp; b&nbsp; = -4<\/p>\n\n\n\n<p>               Given lines are perpendicular <\/p>\n\n\n\n<p>                a + b = 0 <\/p>\n\n\n\n<p>                 k&nbsp; &#8211;&nbsp; 4&nbsp; =&nbsp; 0      <\/p>\n\n\n\n<p>                  \u2234&nbsp; k&nbsp; =&nbsp; 4<\/p>\n\n\n\n<p class=\"has-text-align-center\">           <strong><u>Part \u2013 B<\/u><\/strong>  <\/p>\n\n\n\n<p>1.&nbsp;&nbsp;&nbsp;   Find the angle between the lines  \u221a3&nbsp;x + y = 1 &nbsp;and &nbsp;x +  \u221a3y  = 1 . <\/p>\n\n\n\n<p> Sol:&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;     \u221a3&nbsp;x + y = 1 &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;      (1) <\/p>\n\n\n\n<p>                   x +  \u221a3y  = 1 &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(2) <\/p>\n\n\n\n<p>                 Slope of the line (1) = m<sub>1<\/sub> =&nbsp; &nbsp;&nbsp;&#8211; coefficient of&nbsp; x \/&nbsp; coefficient of y <\/p>\n\n\n\n<p>                                                            =&nbsp;&nbsp; &#8211; \u221a3&nbsp;&nbsp; \/&nbsp;  1 <\/p>\n\n\n\n<p>                 Slope of the line (2) = m<sub>2<\/sub> =&nbsp;&nbsp;&nbsp; &#8211; coefficient of&nbsp; x \/&nbsp; coefficient of y <\/p>\n\n\n\n<p>                                                             =&nbsp;&nbsp; &#8211; 1&nbsp; \/&nbsp;  \u221a3&nbsp; <\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[W.K.T \\ tan\\ \\theta\\ = \\displaystyle\\left\\lvert\\frac{m_1-m_2}{1 + m_1m_2} \\right\\rvert\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ tan\\ \\theta\\ = \\displaystyle\\left\\lvert\\frac{\\frac{-\\sqrt{3}}{1} + \\frac{1} {\\sqrt{3}}}{1 + \\frac{-\\sqrt{3}}{1}\\frac{1} {\\sqrt{3}}} \\right\\rvert\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ tan\\ \\theta\\ = \\displaystyle\\left\\lvert\\frac{-3 + 1}{2\\sqrt{3}}\\right\\rvert\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ tan\\ \\theta\\ = \\displaystyle\\left\\lvert\\frac{-2}{2\\sqrt{3}}\\right\\rvert\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ tan\\ \\theta\\ = \\displaystyle\\left\\lvert\\frac{-1}{\\sqrt{3}}\\right\\rvert\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ tan\\ \\theta\\ = \\frac{1}{\\sqrt{3}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<p>2.&nbsp; &nbsp;&nbsp;Find the equation of the straight line passing through (3,5)&nbsp; and  parallel to x &#8211; 2y -7 = 0.         <\/p>\n\n\n\n<p>Soln:&nbsp;&nbsp; Let the equation of line parallel to &nbsp; x &#8211; 2y -7  = 0 &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;            (1)             <\/p>\n\n\n\n<p>          is x -2y+k = 0 &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;        (2)   <\/p>\n\n\n\n<p>         Equation (2) passes through (3 ,5)<\/p>\n\n\n\n<p>          &nbsp;put x=3, y = 5 in equation (2) <\/p>\n\n\n\n<p>         3 &#8211; 2(5) +k = 0  <\/p>\n\n\n\n<p>          3 &#8211; 10 + k = 0<\/p>\n\n\n\n<p>           k &#8211; 7 = 0<\/p>\n\n\n\n<p>           k = 7<\/p>\n\n\n\n<p>            \u2234 Required line is x &#8211; 2y + 7 = 0 <\/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<p>3.&nbsp;&nbsp;&nbsp;&nbsp;Find the equation of the &nbsp;line passing through (2, 3)&nbsp; and   perpendicular &nbsp;to 4x &#8211; 3y = 10 .   <\/p>\n\n\n\n<p> Soln:&nbsp;&nbsp; Let the equation of line perpendicular to &nbsp;4x &#8211; 3y  = 10 &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;              (1) <\/p>\n\n\n\n<p>              is  \u2013 3x&nbsp; &#8211; 4y + k = 0 &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;(2) <\/p>\n\n\n\n<p>            Equation (2) passes through (2, 3)    <\/p>\n\n\n\n<p>               put x = 2, y = 3 &nbsp;in equation (2) <\/p>\n\n\n\n<p>             -3 (2) &#8211; 4(3) +k = 0 <\/p>\n\n\n\n<p>                -6 &#8211; 12 + k = 0     <\/p>\n\n\n\n<p>                    k = 18<\/p>\n\n\n\n<p>                \u2013 3x&nbsp; &#8211; 4y + 18 = 0   <\/p>\n\n\n\n<p>               &nbsp; \u2234 Required line is 3x +4y  + 18   = 0                                 <\/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<p><a href=\"https:\/\/clnk.in\/qfCP\">https:\/\/clnk.in\/qfCP<\/a><\/p>\n\n\n\n<p class=\"has-text-align-center\"> <strong><u>Part \u2013 C<\/u><\/strong>  <\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><span style=\"font-size: revert; color: initial;\">Show that the equation&nbsp; 2x<\/span><sup style=\"color: initial;\">2<\/sup><span style=\"font-size: revert; color: initial;\">&nbsp; &#8211;&nbsp; 7xy&nbsp; + 3y<\/span><sup style=\"color: initial;\">2<\/sup><span style=\"font-size: revert; color: initial;\"> + 5x &#8211; 5y + 2= 0&nbsp; represents a pair of&nbsp;straight lines.<\/span> <\/li>\n<\/ol>\n\n\n\n<p> Soln:&nbsp; Given&nbsp; 2x<sup>2<\/sup>&nbsp; &#8211;&nbsp; 7xy&nbsp; + 3y<sup>2<\/sup> + 5x &#8211; 5y + 2= 0 <\/p>\n\n\n\n<p>            Comparing with &nbsp;ax<sup>2<\/sup>&nbsp; +&nbsp; 2hxy&nbsp; + by<sup>2<\/sup> + 2gx + 2fy + c = 0 <\/p>\n\n\n\n<p>              a = 2,&nbsp;&nbsp; 2h = &#8211; 7&nbsp;&nbsp;   ,&nbsp; b = 3,&nbsp;&nbsp;&nbsp; 2g = 5&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; ,&nbsp;&nbsp;&nbsp; 2f =- 5&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; ,&nbsp;&nbsp; c = 2 <\/p>\n\n\n\n<p>                           h = &#8211; 7\/ 2&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;g = 5\/ 2&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp; f = &#8211; 5\/ 2  <\/p>\n\n\n\n<p>               To show&nbsp; <span style=\"text-decoration: underline;\">abc&nbsp; +&nbsp; 2fgh \u2013 af<sup>2<\/sup> \u2013 bg<sup>2<\/sup> \u2013 ch<sup>2<\/sup> = 0<\/span> <\/p>\n\n\n\n<p>            abc&nbsp; +&nbsp; 2fgh \u2013 af<sup>2<\/sup> \u2013 bg<sup>2<\/sup> \u2013 ch<sup>2<\/sup> =&nbsp; <\/p>\n\n\n\n<p>              (2) (3) (2)&nbsp; + 2 (-5\/2) (5\/2) (-7\/2) \u2013 2 (-5\/ 2)<sup>2<\/sup>&nbsp; &#8211; 3 (5\/ 2)<sup>2<\/sup>&nbsp; &#8211; 2 (-7\/ 2)<sup>2<\/sup> <\/p>\n\n\n\n<p>              =  12&nbsp; + &nbsp;175\/4&nbsp;&nbsp; &#8211;&nbsp; 50\/4&nbsp; &#8211; 75\/4&nbsp; &#8211; 98\/4 <\/p>\n\n\n\n<p>              =  12&nbsp; +&nbsp; ( 175 \u2013 50 \u2013 75\u2013 98)\/4 <\/p>\n\n\n\n<p>               =&nbsp;&nbsp; 12 + ( 175 \u2013 223)\/4 <\/p>\n\n\n\n<p>                =&nbsp; 12&nbsp; +&nbsp;&nbsp; (&nbsp; -48 \/4 ) <\/p>\n\n\n\n<p>                =&nbsp;&nbsp; 12&nbsp; &#8211; 12&nbsp; =&nbsp; 0 <\/p>\n\n\n\n<p>             Hence, the given equation represents pair of straight lines. <\/p>\n\n\n\n<p>2.&nbsp;&nbsp;Find K if&nbsp;&nbsp; 3x<sup>2<\/sup>&nbsp;+ 7xy&nbsp; + ky<sup>2<\/sup> &#8211; 4x &#8211; 13y &#8211; 7= 0&nbsp; represents a pair of&nbsp; straight lines. &nbsp;  <\/p>\n\n\n\n<p>Soln:    Given&nbsp;  3x<sup>2<\/sup>&nbsp;+ 7xy&nbsp; + ky<sup>2<\/sup> &#8211; 4x &#8211; 13y &#8211; 7= 0  &#8212;&#8212;&#8212;&#8212;&#8211; ( 1 ) <\/p>\n\n\n\n<p>            Comparing with&nbsp; ax<sup>2<\/sup>&nbsp; +&nbsp; 2hxy&nbsp; + by<sup>2<\/sup> + 2gx + 2fy + c = 0&nbsp; <\/p>\n\n\n\n<p>           a = 3,&nbsp;&nbsp; 2h =  7&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; ,&nbsp; b = k,&nbsp;&nbsp;&nbsp; 2g = &#8211; 4&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; ,&nbsp;&nbsp;&nbsp; 2f = &#8211; 13&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; ,&nbsp;&nbsp; c = &#8211; 7<\/p>\n\n\n\n<p>                        h =  7\/ 2&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;      g = &#8211; 2&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp; f = &nbsp;&#8211; 13\/ 2<\/p>\n\n\n\n<p>           Given equation ( 1 ) represents a pair of straight lines <\/p>\n\n\n\n<p>             i.e&nbsp;&nbsp;&nbsp;&nbsp; abc&nbsp; +&nbsp; 2fgh \u2013 af<sup>2<\/sup> \u2013 bg<sup>2<\/sup> \u2013 ch<sup>2<\/sup> =&nbsp;&nbsp; 0 <\/p>\n\n\n\n<p>            (3) (k) (-7)&nbsp; + 2 (- 13\/2) (-2) (7\/2) \u2013 3 (- 13\/ 2)<sup>2<\/sup>&nbsp; &#8211; k (-2)<sup>2<\/sup>&nbsp; + 7 ( 7\/ 2)<sup>2<\/sup> = 0  <\/p>\n\n\n\n<p>             -21k&nbsp; + 91&nbsp;&nbsp; &#8211;&nbsp; 507\/4&nbsp; -4 k&nbsp; + 343\/4 &nbsp;&nbsp;=&nbsp; 0 <\/p>\n\n\n\n<p>              (-84 k&nbsp; + 364 \u2013 507 &#8211; 16k + 343 )\/4&nbsp; =&nbsp; 0<\/p>\n\n\n\n<p>              -100k + 707 &#8211; 507 = 0<\/p>\n\n\n\n<p>              -100k + 200 = 0<\/p>\n\n\n\n<p>                 &nbsp; k&nbsp; =&nbsp; 2 <\/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\" 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<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><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Part \u2013 A 1. &nbsp;Find the perpendicular distance from the point (3, -5) to the straight line 3x-4y-26=0. Soln: W.K.T&nbsp; The length of the perpendicular distance from (x1,y1) to the line ax + by + c = 0&nbsp; is \u00b1&nbsp; (ax1 + by1 + c)\/ \u221a(a2 + b2 ) Given straight line is &nbsp;&nbsp;&nbsp; 3x-4y-26 [&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":[1],"tags":[],"class_list":["post-14590","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>1.1 &#8211; N &#8211; Analytical Geometry Exercise Problems With Solutions - 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=14590\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"1.1 &#8211; N &#8211; Analytical Geometry Exercise Problems With Solutions - YANAMTAKSHASHILA\" \/>\n<meta property=\"og:description\" content=\"Part \u2013 A 1. &nbsp;Find the perpendicular distance from the point (3, -5) to the straight line 3x-4y-26=0. Soln: W.K.T&nbsp; The length of the perpendicular distance from (x1,y1) to the line ax + by + c = 0&nbsp; is \u00b1&nbsp; (ax1 + by1 + c)\/ \u221a(a2 + b2 ) Given straight line is &nbsp;&nbsp;&nbsp; 3x-4y-26 [&hellip;]\" \/>\n<meta property=\"og:url\" content=\"https:\/\/yanamtakshashila.com\/?p=14590\" \/>\n<meta property=\"og:site_name\" content=\"YANAMTAKSHASHILA\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/profile.php?id=100063680185552\" \/>\n<meta property=\"article:author\" content=\"https:\/\/www.facebook.com\/profile.php?id=100063680185552\" \/>\n<meta property=\"article:published_time\" content=\"2021-05-09T06:24:44+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2024-04-13T10:36:19+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/ad.admitad.com\/b\/87u3svniba109003a6887d95a12660\/\" \/>\n<meta name=\"author\" content=\"rajuviswa\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"rajuviswa\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"11 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/?p=14590#article\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/?p=14590\"},\"author\":{\"name\":\"rajuviswa\",\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/#\\\/schema\\\/person\\\/a990a0af264ac2298c19fa61d2bda16e\"},\"headline\":\"1.1 &#8211; N &#8211; Analytical Geometry Exercise Problems With Solutions\",\"datePublished\":\"2021-05-09T06:24:44+00:00\",\"dateModified\":\"2024-04-13T10:36:19+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/?p=14590\"},\"wordCount\":2209,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/#\\\/schema\\\/person\\\/a990a0af264ac2298c19fa61d2bda16e\"},\"image\":{\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/?p=14590#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/ad.admitad.com\\\/b\\\/87u3svniba109003a6887d95a12660\\\/\",\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\\\/\\\/yanamtakshashila.com\\\/?p=14590#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/?p=14590\",\"url\":\"https:\\\/\\\/yanamtakshashila.com\\\/?p=14590\",\"name\":\"1.1 &#8211; N &#8211; Analytical Geometry Exercise Problems With Solutions - YANAMTAKSHASHILA\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/#website\"},\"primaryImageOfPage\":{\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/?p=14590#primaryimage\"},\"image\":{\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/?p=14590#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/ad.admitad.com\\\/b\\\/87u3svniba109003a6887d95a12660\\\/\",\"datePublished\":\"2021-05-09T06:24:44+00:00\",\"dateModified\":\"2024-04-13T10:36:19+00:00\",\"breadcrumb\":{\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/?p=14590#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\\\/\\\/yanamtakshashila.com\\\/?p=14590\"]}]},{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/?p=14590#primaryimage\",\"url\":\"https:\\\/\\\/ad.admitad.com\\\/b\\\/87u3svniba109003a6887d95a12660\\\/\",\"contentUrl\":\"https:\\\/\\\/ad.admitad.com\\\/b\\\/87u3svniba109003a6887d95a12660\\\/\"},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/?p=14590#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\\\/\\\/yanamtakshashila.com\\\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"1.1 &#8211; N &#8211; Analytical Geometry Exercise Problems With Solutions\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/#website\",\"url\":\"https:\\\/\\\/yanamtakshashila.com\\\/\",\"name\":\"yanamtakshashila.com\",\"description\":\"one stop solutions\",\"publisher\":{\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/#\\\/schema\\\/person\\\/a990a0af264ac2298c19fa61d2bda16e\"},\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\\\/\\\/yanamtakshashila.com\\\/?s={search_term_string}\"},\"query-input\":{\"@type\":\"PropertyValueSpecification\",\"valueRequired\":true,\"valueName\":\"search_term_string\"}}],\"inLanguage\":\"en-US\"},{\"@type\":[\"Person\",\"Organization\"],\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/#\\\/schema\\\/person\\\/a990a0af264ac2298c19fa61d2bda16e\",\"name\":\"rajuviswa\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\\\/\\\/i0.wp.com\\\/yanamtakshashila.com\\\/wp-content\\\/uploads\\\/2024\\\/12\\\/LOGO-PNG.png?fit=3600%2C3600&ssl=1\",\"url\":\"https:\\\/\\\/i0.wp.com\\\/yanamtakshashila.com\\\/wp-content\\\/uploads\\\/2024\\\/12\\\/LOGO-PNG.png?fit=3600%2C3600&ssl=1\",\"contentUrl\":\"https:\\\/\\\/i0.wp.com\\\/yanamtakshashila.com\\\/wp-content\\\/uploads\\\/2024\\\/12\\\/LOGO-PNG.png?fit=3600%2C3600&ssl=1\",\"width\":3600,\"height\":3600,\"caption\":\"rajuviswa\"},\"logo\":{\"@id\":\"https:\\\/\\\/i0.wp.com\\\/yanamtakshashila.com\\\/wp-content\\\/uploads\\\/2024\\\/12\\\/LOGO-PNG.png?fit=3600%2C3600&ssl=1\"},\"sameAs\":[\"http:\\\/\\\/yanamtakshashila.wordpress.com\",\"https:\\\/\\\/www.facebook.com\\\/profile.php?id=100063680185552\",\"https:\\\/\\\/www.instagram.com\\\/rajuviswa\\\/?hl=en\",\"https:\\\/\\\/www.youtube.com\\\/channel\\\/UCjJ2KWWvsFm6F42UtMdbxzw\"],\"url\":\"https:\\\/\\\/yanamtakshashila.com\\\/?author=187055548\"}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"1.1 &#8211; N &#8211; Analytical Geometry Exercise Problems With Solutions - YANAMTAKSHASHILA","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/yanamtakshashila.com\/?p=14590","og_locale":"en_US","og_type":"article","og_title":"1.1 &#8211; N &#8211; Analytical Geometry Exercise Problems With Solutions - YANAMTAKSHASHILA","og_description":"Part \u2013 A 1. &nbsp;Find the perpendicular distance from the point (3, -5) to the straight line 3x-4y-26=0. Soln: W.K.T&nbsp; The length of the perpendicular distance from (x1,y1) to the line ax + by + c = 0&nbsp; is \u00b1&nbsp; (ax1 + by1 + c)\/ \u221a(a2 + b2 ) Given straight line is &nbsp;&nbsp;&nbsp; 3x-4y-26 [&hellip;]","og_url":"https:\/\/yanamtakshashila.com\/?p=14590","og_site_name":"YANAMTAKSHASHILA","article_publisher":"https:\/\/www.facebook.com\/profile.php?id=100063680185552","article_author":"https:\/\/www.facebook.com\/profile.php?id=100063680185552","article_published_time":"2021-05-09T06:24:44+00:00","article_modified_time":"2024-04-13T10:36:19+00:00","og_image":[{"url":"https:\/\/ad.admitad.com\/b\/87u3svniba109003a6887d95a12660\/","type":"","width":"","height":""}],"author":"rajuviswa","twitter_card":"summary_large_image","twitter_misc":{"Written by":"rajuviswa","Est. reading time":"11 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/yanamtakshashila.com\/?p=14590#article","isPartOf":{"@id":"https:\/\/yanamtakshashila.com\/?p=14590"},"author":{"name":"rajuviswa","@id":"https:\/\/yanamtakshashila.com\/#\/schema\/person\/a990a0af264ac2298c19fa61d2bda16e"},"headline":"1.1 &#8211; N &#8211; Analytical Geometry Exercise Problems With Solutions","datePublished":"2021-05-09T06:24:44+00:00","dateModified":"2024-04-13T10:36:19+00:00","mainEntityOfPage":{"@id":"https:\/\/yanamtakshashila.com\/?p=14590"},"wordCount":2209,"commentCount":0,"publisher":{"@id":"https:\/\/yanamtakshashila.com\/#\/schema\/person\/a990a0af264ac2298c19fa61d2bda16e"},"image":{"@id":"https:\/\/yanamtakshashila.com\/?p=14590#primaryimage"},"thumbnailUrl":"https:\/\/ad.admitad.com\/b\/87u3svniba109003a6887d95a12660\/","inLanguage":"en-US","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/yanamtakshashila.com\/?p=14590#respond"]}]},{"@type":"WebPage","@id":"https:\/\/yanamtakshashila.com\/?p=14590","url":"https:\/\/yanamtakshashila.com\/?p=14590","name":"1.1 &#8211; N &#8211; Analytical Geometry Exercise Problems With Solutions - YANAMTAKSHASHILA","isPartOf":{"@id":"https:\/\/yanamtakshashila.com\/#website"},"primaryImageOfPage":{"@id":"https:\/\/yanamtakshashila.com\/?p=14590#primaryimage"},"image":{"@id":"https:\/\/yanamtakshashila.com\/?p=14590#primaryimage"},"thumbnailUrl":"https:\/\/ad.admitad.com\/b\/87u3svniba109003a6887d95a12660\/","datePublished":"2021-05-09T06:24:44+00:00","dateModified":"2024-04-13T10:36:19+00:00","breadcrumb":{"@id":"https:\/\/yanamtakshashila.com\/?p=14590#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/yanamtakshashila.com\/?p=14590"]}]},{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/yanamtakshashila.com\/?p=14590#primaryimage","url":"https:\/\/ad.admitad.com\/b\/87u3svniba109003a6887d95a12660\/","contentUrl":"https:\/\/ad.admitad.com\/b\/87u3svniba109003a6887d95a12660\/"},{"@type":"BreadcrumbList","@id":"https:\/\/yanamtakshashila.com\/?p=14590#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/yanamtakshashila.com\/"},{"@type":"ListItem","position":2,"name":"1.1 &#8211; N &#8211; Analytical Geometry Exercise Problems With Solutions"}]},{"@type":"WebSite","@id":"https:\/\/yanamtakshashila.com\/#website","url":"https:\/\/yanamtakshashila.com\/","name":"yanamtakshashila.com","description":"one stop solutions","publisher":{"@id":"https:\/\/yanamtakshashila.com\/#\/schema\/person\/a990a0af264ac2298c19fa61d2bda16e"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/yanamtakshashila.com\/?s={search_term_string}"},"query-input":{"@type":"PropertyValueSpecification","valueRequired":true,"valueName":"search_term_string"}}],"inLanguage":"en-US"},{"@type":["Person","Organization"],"@id":"https:\/\/yanamtakshashila.com\/#\/schema\/person\/a990a0af264ac2298c19fa61d2bda16e","name":"rajuviswa","image":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2024\/12\/LOGO-PNG.png?fit=3600%2C3600&ssl=1","url":"https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2024\/12\/LOGO-PNG.png?fit=3600%2C3600&ssl=1","contentUrl":"https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2024\/12\/LOGO-PNG.png?fit=3600%2C3600&ssl=1","width":3600,"height":3600,"caption":"rajuviswa"},"logo":{"@id":"https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2024\/12\/LOGO-PNG.png?fit=3600%2C3600&ssl=1"},"sameAs":["http:\/\/yanamtakshashila.wordpress.com","https:\/\/www.facebook.com\/profile.php?id=100063680185552","https:\/\/www.instagram.com\/rajuviswa\/?hl=en","https:\/\/www.youtube.com\/channel\/UCjJ2KWWvsFm6F42UtMdbxzw"],"url":"https:\/\/yanamtakshashila.com\/?author=187055548"}]}},"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_likes_enabled":true,"jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/pc3kmt-3Nk","_links":{"self":[{"href":"https:\/\/yanamtakshashila.com\/index.php?rest_route=\/wp\/v2\/posts\/14590","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/yanamtakshashila.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/yanamtakshashila.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/yanamtakshashila.com\/index.php?rest_route=\/wp\/v2\/users\/187055548"}],"replies":[{"embeddable":true,"href":"https:\/\/yanamtakshashila.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=14590"}],"version-history":[{"count":68,"href":"https:\/\/yanamtakshashila.com\/index.php?rest_route=\/wp\/v2\/posts\/14590\/revisions"}],"predecessor-version":[{"id":45618,"href":"https:\/\/yanamtakshashila.com\/index.php?rest_route=\/wp\/v2\/posts\/14590\/revisions\/45618"}],"wp:attachment":[{"href":"https:\/\/yanamtakshashila.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=14590"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/yanamtakshashila.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=14590"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/yanamtakshashila.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=14590"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}