Question

From a point perpendicular tangents are drawn to ellipse $x^2+2y^2=2$. The chord of contact touches a circle which is concentric with given ellipse. Then find the ratio of maximum and minimum area of circle ___

Answer 1

The director circle of ellipse $\frac{x^2}{2}+\frac{y^2}{1}=1$ is $x^2+y^2=2+1=3$

Let the point on ellipse be $P(\sqrt{3}cos\theta ,\sqrt{3}sin\theta )$

Equation of chord of contact is $x{x}_1+2y{y}_1=2$

$x.\sqrt{3}\cos \theta +2y\sqrt{3}\sin \theta -2=0$

It touches $x^2+y^2=r^2$

$r=\frac{2}{\sqrt{3co{s}^2\theta +12si{n}^2\theta }}=\frac{2}{\sqrt{3+9si{n}^2\theta }}$
When $sin\theta =0$ then r attains maximum value,
So, $r_{max}=\frac{2}{\sqrt{3}}$
When $\sin \theta =1$ then r attains minimum value,
So, $r_{min}=\frac{2}{\sqrt{12}}=\frac{2}{2\sqrt{3}}=\frac{1}{\sqrt{3}}$
$\Rightarrow \frac{r_{max}}{r_{min}}=\frac{\frac{2}{\sqrt{3}}}{\frac{1}{\sqrt{3}}}=2$
Therefore, the ratio of maximum and minimum area of circle $=(\frac{r_{max}}{r_{min}}{)}^2=4$