Mathematics - Circle Question with Solution | TestHub

$\mathrm{Let\; P\; be }(\mathrm{cos\; \theta },\mathrm{sin\; \theta })$, where
$\mathrm{\theta }\ne 0,\mathrm{\pi }$
Tangent at$P :x\cos \theta +y\sin \theta =1$........(i)
Tangent at$S :x=1$.........(ii)
$\therefore$By (i) and (ii) :$Q\left(1,\frac{1-\cos \theta }{\sin \theta }\right)$
Line through Q parallel to RS :
$y=\frac{1-\cos \theta }{\sin \theta } \Rightarrow y=\tan \frac{\theta }{2}$..........(iii)
Normal at$P :y=\frac{\sin \theta }{\cos \theta }x \Rightarrow y=\tan \theta .x$$\Rightarrow y=\left(\frac{2t\mathit{an}\frac{\theta }{2}}{1-t{\mathit{an}}^{\mathit{2}}\frac{\theta }{2}}\right)x$........(iv)
Point of intersection of equation (iii) and (iv),$E :h=\frac{1-{\tan }^2\frac{\theta }{2}}{2};k=\tan \frac{\theta }{2}$
Eliminating$\theta :h=\frac{1-k^2}{2} \Rightarrow y^2=1-2x$
$\left(\frac{1}{3},\frac{1}{\sqrt{3}}\right) \& \left(\frac{1}{3},-\frac{1}{\sqrt{3}}\right)$satisfy the locus.