Question

If the chord of contact of tangents from a point on the circle $x^2+y^2=a^2$ to the circle $x^2+y^2=b^2$ touches the circle $x^2+y^2=c^2$, then $a,b,c$ are in

• A. A.P.
• B. G.P.
• C. H.P.
• D. none of these

Answer 1

The correct option is B G.P.

Let $(x_1,y_1)$ be any point on the circle $x^2+y^2=a^2$

then, ${x_1}^2+{y_1}^2=a^2$ ...(1)

Equation of chord of contact of tangents from $(x_1,y_1)$ to the circle is $x{x}_1+y{y}_1=b^2$ ...(2)

$\because$ (2) touches the circle $x^2+y^2=c^2$

$\therefore$ the length of $\perp$ from center $(0,0)$ on (2) is $=$ radius $=c$

$\Rightarrow \frac{b^2}{\sqrt{{x_1}^2+{y_1}^2}}=c\Rightarrow \frac{b^2}{\sqrt{a^2}}=c\Rightarrow b^2=ac$

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