પરવલય પર બિંદુ $P(3sec\ \theta , 2tan\ \theta)$ લો.

સ્પર્શકનું સમીકરણ $\frac{{{\rm{x}}\,{\rm{sec}}\theta }}{{\rm{3}}}\,\, – \,\,\frac{{y\,\tan \,\theta }}{2}\,\, = \,1\,$

$\,|p|\,\, = \,\,r\,\,\,\,\,\,\,\,\frac{{\left| {\frac{4}{3}\,\,\sec \,\,\theta \,\, – \,\,1} \right|}}{{\sqrt {\frac{{{{\sec }^2}\theta }}{9}\, + \;\,\frac{{{{\tan }^2}\theta }}{4}} }}\,\, = \,\,4$

$ \Rightarrow \,\,\frac{{16}}{9}\,\,{\sec ^2}\theta \,\, + \;\,1\,\, – \,\,\frac{8}{3}\,\,\sec \theta \,\, = \,16\,\,\left( {\frac{{4\,{{\sec }^2}\theta \,\, + \;\,9\,\,{{\tan }^2}\,\,\theta }}{{4\,\, \times \,\,9}}} \right)$

$16\,{\sec ^2}\theta \,\, + \;\,9\,\, – \,\,24\,\sec \,\theta \,\, = \,\,52\,\,{\sec ^2}\theta \,\, – \,\,36$

$ \Rightarrow 36\,\,{\sec ^2}\theta \,\, + \,24\,\,\sec \,\theta \,\, – \,\,45\,\, = \,\,0$

$ \Rightarrow \,\,12\,\,{\sec ^2}\,\,\theta \,\, + \;\,8\,\,\sec \,\theta \,\, – \,\,15\,\, = \,\,0$

$\begin{array}{l}
 \Rightarrow \,\,12\,\,{\sec ^2}\,\,\theta \,\, + \;\,18\,\,\sec \theta \,\, – \,\,10\,\,\sec \,\,\theta \,\, – \,\,15\,\, = \,\,0\\
 \Rightarrow \,\,\left( {6\,\,\sec \,\,\theta \,\, – \,\,5} \right)\,\,\left( {2\,\,\sec \theta \,\, + \,\,3} \right)\,\, = \,\,0
\end{array}$

$\sec \theta \,\, = \,\,\frac{5}{6}\,$ (શકય નથી) $,\,\,\,\,\sec \,\,\theta \,\, = \,\, – \frac{3}{2}$

$\tan \,\,\theta \,\, = \,\, \pm \,\,\sqrt {\frac{9}{4}\,\, – \,\,1} \,\, = \,\, \pm \,\,\frac{{\sqrt 5 }}{2}\,\,\,\,$

(ઢાળ ધન છે.${\, \Rightarrow \,\,\tan \,\,\theta \,\, = \,\, – \frac{{\sqrt 5 }}{2}}$)

આથી માંગેલું સમીકરણ $ – \frac{{3x}}{{2\,\, \times \,\,3}}\,\,\, + \,\,\frac{{y\sqrt 5 }}{{2\,\, \times \,\,2}}\,\, = \,\,1\,\,\, \Rightarrow \,\,2x\,\, – \,\,\sqrt 5 y\,\, + \;\,4\,\, = \,\,0$