Question

lf the polars of the points on the circle $x^2+y^2=a^2$ with respect to the circle $x^2+y^2=c^2$ touches the circle $x^2+y^2=b^2$ then

• A. $a,b,c$ are in G.P
• B. $b,a,c$ are in A.P
• C. $c=\sqrt{ab}$
• D. $c=\frac{2ab}{a+b}$

Answer 1

The correct option is C $c=\sqrt{ab}$

Let $P(a\cos \theta ,a\sin \theta )$ be point on circle $x^2+y^2=a^2$.
The equation of polars from $P(a\cos \theta ,a\sin \theta )$ with respect to
circle $x^2+y^2=c^2$ is
$xa\cos \theta +ya\sin \theta =c^2$
$x\cos \theta +y\sin \theta =\frac{c^2}{a}-------\left(1\right)$
Then $x\cos \theta +y\sin \theta -\frac{c^2}{a}=0$ is a tangent to
circle $x^2+y^2=b^2$
$\therefore b=\left|\frac{\frac{c^2}{a}}{\sqrt{{\cos }^{2}a+{\sin }^{2}a}}\right|$
$b=\frac{c^2}{a}$
$\therefore c^2=ab$
$\Rightarrow c=\sqrt{ab}$