$P\,\,\left( {a\,\sec \,\,\theta ,\,\,b\,\,\tan \,\,\theta } \right)$ અને $Q\,\,\left( {a\,\,\sec \varphi ,\,\,b\,\,\tan \,\,\varphi } \right)$ આપેલા છે 

$P$ બિંદુ આગળના સ્પર્શકનું સમીકરણ $:\,\,\,\frac{{x\,\,\sec \,\theta }}{a}\,\, – \,\,\frac{{y\,\tan \,\,\theta }}{b}\,\, = \,\,1\,$

સ્પર્શક ઢાળ $m\,\, = \,\,\frac{b}{{\tan \,\,\theta }}\,\, \times \,\,\frac{{\sec \theta }}{a}\,\, = \,\,\frac{b}{a}\,\,.\,\,\frac{1}{{\sin \,\,\theta }}$

આથી $P$ આગળના લંબનું સમીકરણ $:\,\,y\,\, – \,\,b\,\tan \,\,\theta \,\, = \,\, – \frac{{a\,\sin \,\,\theta }}{b}\,\,\left( {x\,\, – \,\,a\,\sec \theta } \right)$

અથવા $by\,\, – \,\,{b^2}\,\,\tan \,\,\theta \,\, = \,\, – a\sin \,\,\theta \,\,x\,\, + \;\,{a^2}\tan \,\theta $

અથવા $a\,\sin \theta x\,\, + \,\,by\,\, = \,\,\left( {{a^2}\,\, + \;\,{b^2}} \right)\,\,\tan \,\,\theta \,\,………\left( i \right)$

તે જ રીતે $Q$ આગળનું લંબનું સમીકરણ $:\,\,a\sin \,\varphi \,\, + \,by\,\, = \,\,\left( {{a^2}\,\, + \;\,{b^2}} \right)\,\,\tan \,\varphi \,\,\,\,……..\left( {ii} \right)$

$\left( i \right)$ નો $\,\sin \,\,\varphi $ અને $\left( {ii} \right)$ નો $\sin \,\,\theta \,$ દ્વારા ગુણોતર કરતાં 

$a\,\sin \,\theta \,\sin \,\varphi x\,\, + \;\,b\,\sin \,\varphi y\,\, = \,\,\left( {{a^2}\,\, + \;\,{b^2}} \right)\,\,\tan \,\theta \,\sin \,\varphi $

$a\,\sin \,\varphi \,\sin \,\theta \,x\,\, + \;\,b\sin \,\theta y\,\, = \,\,\left( {{a^2}\,\, + \;\,{b^2}} \right)\,\,\tan \,\varphi \,\sin \,\theta $

કિંમતો મૂકતાં $\left( {\sin \,\,\varphi \,\, – \,\,\sin \,\,\theta } \right)\,\, = \,\,\left( {{a^2}\,\, + \,\,{b^2}} \right)\,\,\left( {\tan \,\theta \,\,\sin \,\,\varphi \,\, – \,\,\tan \,\varphi \,\,\sin \,\theta } \right)$

$\therefore \,\,y\,\, = \,\,k\,\, = \,\,\frac{{{a^2}\,\, + \;\,{b^2}}}{b}\,\,.\,\,\frac{{\tan \,\,\theta \,\,\sin \,\,\varphi \,\, – \,\,\tan \,\,\varphi \,\,\sin \,\,\theta }}{{\sin \,\,\varphi \,\, – \,\,\sin \,\,\theta }}$

$\because \,\,\theta \,\, + \;\,\varphi \,\, = \,\,\frac{\pi }{2}\,\, \Rightarrow \,\,\varphi \,\, = \,\,\frac{\pi }{2}\,\, – \,\,\theta \,$

$ \Rightarrow \,\,\sin \,\,\varphi \,\, = \,\,\cos \,\,\theta \,$ અને $\tan \,\,\varphi \,\, = \,\,\cot \,\,\theta $

$\therefore \,\,y\,\, = \,\,k\,\, = \,\,\frac{{{a^2}\,\, + \;\,{b^2}}}{b}\,\,.\,\,\frac{{\tan \,\,\theta \,\,\cos \,\,\theta \,\, – \,\,\cot \,\,\theta \,\,\sin \,\,\theta }}{{\cos \,\,\theta \, – \,\,\sin \,\,\theta }}$

$ \Rightarrow \,\,\frac{{{a^2}\,\, + \;\,{b^2}}}{b}\,\,\left( {\frac{{\sin \,\,\theta \,\, – \,\,\cos \,\,\theta }}{{\cos \,\,\theta \,\, – \,\,\sin \,\,\theta }}} \right)\,\, = \,\, – \,\,\frac{{\left( {{a^2}\,\, + \;\,{b^2}} \right)}}{b}$