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જો સુરેખા $\,x\cos \,\,\alpha \,\, + \,\,y\,\sin \,\,\alpha \,\, = \,\,p$ એ અતિવલય
$\frac{{{x^2}}}{{{a^2}}}\,\, - \,\,\frac{{{y^2}}}{{{b^2}}}\,\, = \,\,1\,$ નો સ્પર્શક હોય , તો .....
$a^2cos^2 \alpha + b^2 sin^2 \alpha = p^2$
$a^2 cos^2 \alpha - b^2 sin^2 \alpha = p^2$
$a^2 sin^2 \alpha + b^2 cos^2 \alpha = p^2$
$a^2 sin^2 \alpha - b^2 cos^2 \alpha = p^2$
Solution
$x\,\,\cos \alpha \,\, + \;\,y\sin \,\,\alpha \,\, = \,\,p\,\,\, \Rightarrow \,\,y\,\, = \,\, – \cot \alpha \,\,x\,\, + \;\,p\,\cos ec\,\,\alpha \,$
તે અતિવલય $\,\,\frac{{{x^2}}}{{{a^2}}}\,\, – \,\,\frac{{{y^2}}}{{{b^2}}}\,\, = \,\,1$ નો સ્પર્શક છે
તેથી ${p^2}\,\cos e{c^2}\alpha \,\, = \,\,{a^2}{\cot ^2}\alpha \,\, – \,\,{b^2}$
$ \Rightarrow \,\,{a^2}{\cos ^2}\alpha \,\, – \,\,{b^2}\,{\sin ^2}\,\alpha \,\, = \,\,{p^2}$
