Question

If the pole of any line with respect to the circle $x^2+y^2=c^2$ lies on the circle $x^2+y^2=9c^2$, then the line will be a tangent to the circle $x^2+y^2=c^2/9$.

Answer 1

Suppose the co-ordinates of the pole are $(x_1,y_2)$ and as it lies on
$x^2+y^2=9c^2\therefore x_1^2+y_1^2=9c^2.....\left(1\right)$
The polar of this point w.r.t. the circle $x^2+y^2=c^2$ is $x{x}_1+y{y}_1=c^2$.
If this polar touches the circle $x^2+y^2=c^2/9$ then perpendicular from centre $(0,0)$ should be equal to radius $c/3$.
$\therefore \frac{0+0-c^2}{\sqrt{(x_1^2+y_1^2)}}=\frac{c}{3}$
or $9c^2=x_1^2+y_1^2$
Above is true y the help of $\left(1\right)$.