Question

If the chords of contact of points on $x^2+y^2=a^2$ with respect to the circle $x^2+y^2=b^2$ touch the circle $x^2+y^2=c^2$, then $a,b,c$ are in

• A. A.P.
• B. G.P.
• C. H.P.
• D. A.G.P.

Answer 1

Answer: (B)

G.P.

$S_1:x^2+y^2=a^2$

$S_2:x^2+y^2=b^2$

$S_3:x^2+y^2=c^2$

Let $P(a\cos \theta ,b\sin \theta )$ be a point on $S_1$
So, chord of contact from $P(a\cos \theta ,b\sin \theta )$ to $S_2$ is $ax\cos \theta +by\sin \theta =b^2$ ...(1)
But, given $ax\cos \theta +by\sin \theta -b^2=0$ touches $S_3$
So, equation of tangent to $S_3$ is
$cx\cos \theta +yc\sin \theta =c^2$
$x\cos \theta +y\sin \theta =c$ ...(2)

Compare (1) & (2), we get

$\frac{a\cos \theta }{\cos \theta }=\frac{b\sin \theta }{\sin \theta }=\frac{b^2}{c}$

$\Rightarrow ac=b^2$ and $b=c$

$a,b,c$ are in G.P.

Hence, option B.