Mathematics - Ellipse Question with Solution | TestHub

Smaller ellipse is $\frac{x^2}{4}+\frac{y^2}{1}=1$ larger is $\frac{x^2}{6}+\frac{\lambda y^2}{6}=1$.

Any tangent to smaller is $\frac{x}{2}\cos \theta +y\sin \theta =1$

Let it meet larger at P and Q . Let tangent at $\mathrm{P},\mathrm{Q}$ intersect at ' A ' $(h,k)$

PQ is chord of contact wrt $\mathrm{A}\therefore PQ=\frac{hx}{6}+\frac{\lambda yk}{6}=1$

Comparing (1) and (2) $h=3\cos \theta k=\frac{6\sin \theta }{\lambda }$ $(h,k)\in x^2+y^2=\frac{6+6}{\lambda }\Rightarrow 9{\cos }^2\theta +\frac{6^2}{{\lambda }^2}{\sin }^2\theta =6+\frac{6}{\lambda }$

$\Rightarrow \lambda =2$