Question

Consider three circles $C_1,C_2$ and $C_3$ such that $C_2$ is the director circle of $C_1$ and $C_3$ is the director circle of $C_2,$ Tangents to $C_1$ from any point on $C_3$ intersect $C_2$ at $P$ and $Q.$ Find the angle between the tangents to $C_2$ at $P$ and $Q.$ Also identify the locus of the point of intersection of tangents at $P$ and $Q$

Answer 1

Let $C_1$ be a circle of radius $r$ So radius of $C_2=\sqrt{2}r$ and that of $C_3=2r$

In $\Delta AOT$

$\sin \frac{\theta }{2}=\frac{r}{2r}=\frac{1}{2}\Rightarrow \theta =\frac{\pi }{3}$

$\Rightarrow \angle ATB=\frac{\pi }{3}$

$\Rightarrow \mathrm{In}\Delta AOP,\sin \angle APO=\frac{r}{r\sqrt{2}}\Rightarrow \angle APO=\frac{\pi }{4}$

In $\Delta OPT\angle POT=\frac{\pi }{4}-\frac{\pi }{6}=\frac{\pi }{12}$

In $\Delta OPH$

$\Rightarrow \angle OPH=\frac{\pi }{2}$

$\angle PHO=\pi -(\frac{\pi }{2}+\frac{\pi }{12})=\frac{5\pi }{12}$

$\angle PHQ=\frac{5\pi }{6}\cdot As\angle POQ=\frac{\pi }{6}$

$OPHQ$ is a cyclic quadrilateral.

Hence, locus of $H$ is a circle.