Question

From a point, perpendicular tangents are drawn to the ellipse $x^2+2y^2=2.$ The chord of contact touches a circle concentric with the given ellipse. The ratio of the maximum, minimum areas of the circle is

Answer 1

Given ellipse equation is $\frac{x^2}{2}+\frac{y^2}{1}=1$
So, Equation of director circle is $x^2+y^2=3$ and any point $P$on circle be $(\sqrt{3}\cos \theta ,\sqrt{3}\sin \theta )$
So,equation of chord of contact for ellipse is:
$\left(\sqrt{3}\cos \theta \right)x+\left(2\sqrt{3}\sin \theta \right)y-2=0$
Let circle equation concentric with given ellipse be $x^2+y^2=r^2$
$\Rightarrow r=\left|\frac{-2}{\sqrt{3{\cos }^2\theta +12{\sin }^2\theta }}\right|$
$\Rightarrow r^2=\frac{4}{3+9{\sin }^2\theta }$
$\therefore$ Required ratio : $\frac{\text{Maximum area of circle}}{\text{Minimum area of circle}}=\frac{\pi \cdot \frac{4}{3}}{\pi \cdot \frac{1}{3}}=4$