From a point, perpendicular tangents are drawn to the ellipse x² + 2y² = 2. The chord of contact touches a circle concentric with the given ellipse. The ratio of the maximum minimum areas of the circle is$\mathrm{\lambda }:1$. Then  $\mathrm{\lambda }$is .........

$$\begin{gathered} Given ellipse x^2+2y^2=2 \\ Now equation of director circle is x^2+y^2=3 \end{gathered}$$

Let$\mathrm{P}(\sqrt{3}\cos \mathrm{\theta },\sqrt{3}\sin \mathrm{\theta })$ be a point on the director circle
Equation of chord of contact of P w.r.t (1) is 
$\sqrt{3}\cos \mathrm{\theta }\cdot \mathrm{x}+2\sqrt{3}\sin \mathrm{\theta }\cdot \mathrm{y}=2$
If it is a tangent to the circle ${\mathrm{x}}^2+{\mathrm{y}}^2=r^2\text{, then }\mathrm{r}=\mathrm{d}$

$r=\frac{\left|-2\right|}{\sqrt{3{\cos }^2\theta +12{\sin }^2\theta }}$
Maximum area of the circle $=\frac{4\mathrm{\pi }}{3}(\text{ at }\mathrm{\theta }=0)$
Minimum area of the circle$=\frac{\mathrm{\pi }}{3}\left(\text{ at }\mathrm{\theta }=\frac{\mathrm{\pi }}{2}\right)$

$Ratio of maximum and minimum areas of circle =\frac{4\pi }{3}:\frac{\pi }{3}$
$$\begin{gathered} \Rightarrow \mathrm{\lambda }:1=4:1 \\ \therefore \mathrm{\lambda }=4 \end{gathered}$$