Question

If the locus of a point which moves so that the line joining the point of contact of the tangents drawn from it to the circle $x^2+y^2=b^2$ touches the circle $x^2+y^2=a^2$, is 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

Answer: (B)

G.P.

Let $P(h,k)$ be any point on the locus. Equation of the chord contact of $P$ with respect to the circle $x^2+y^2=b^2$ is $hx+ky=b^2$.

If it touches the circle $x^2+y^2=a^2$, then

$\left|\frac{-b^2}{\sqrt{h^2+k^2}}\right|=a\Rightarrow a^2(h^2+k^2)=b^4$

So that the locus of $P(h,k)$ is $x^2+y^2={\left(\frac{b^2}{a}\right)}^2$

$\therefore c^2={\left(\frac{b^2}{a}\right)}^2\Rightarrow ac=b^2$

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