Question

Consider two concentric circles $C_1:x^2+y^2=1,C_2:x^2+y^2=4$. Tangent are drawn to $C_1$ from any point $P$ on $C_2$. These tangents again meet circles $C_2$ at $A$ and $B$. Prove that the line joining $A$ and $B$ will touch $C_1$. Further, find the locus of the point of intersection of tangents drawn to $C_2$ at $A$ and $B$.

Answer 1

Radius of $C1=1cm$

Radius of $C2=2cm$

In $△OPM$

$\sin \angle MPO=\frac{1}{2}\angle MPO={30}^o\angle APB=2\angle MPO\angle APB={60}^o$

In $△APB$

$AP=AB\angle PAB=\angle PBA\angle PAB+\angle PBA+\angle APB={180}^{o}2\angle PBA+{60}^o={180}^o\angle PBA=\angle PAB={60}^o$

Hence, $△APB$ is an equilateral triangle

$C1$ is incircle and $C2$ is circumcircle

So, the line $AB$ touches the incircle $C2$

For tangents, at $A$ and $B$ the point of intersection is $S$

$\angle ASB={60}^o$ and $\angle OSA={30}^o$

$\sin {30}^o=\frac{OA}{OB}\Rightarrow OS=2OA=4$

Locus of $S$ with centre $O$ and radius $4$

$x^2+y^2=16$