Question

The chord of contact of tangents drawn from any point on the circle $x^2+y^2=a^2$ to the circle $x^2+y^2=b^2$ touches the circle $x^2+y^2=c^2$. Show that $a,b,c$ are in G.P. $(a>b)$.

Answer 1

$(a\cos \theta ,a\sin \theta )$ is on $x^2+y^2=a^2$.
$\therefore$ Chord of contact is $ax\cos \theta +ay\sin \theta =b^2$
It touches $x^2+y^2=c^2$
$\frac{b^2}{a\sqrt{({\cos }^2\theta +{\sin }^2\theta )}}=c$ or $b^2=ac$
$\therefore a,b,c$ are in G.P.