If a variable tangent of the circle ${\mathrm{x}}^2+{\mathrm{y}}^2=1$ intersect the ellipse ${\mathrm{x}}^2+2{\mathrm{y}}^2=4$ at P and Q then the locus of the points of intersection of the tangents at P and Q is

${\mathrm{x}}^2+{\mathrm{y}}^2=1;{\mathrm{x}}^2+2{\mathrm{y}}^2=4$
Let $\mathrm{R}({\mathrm{x}}_1,{\mathrm{y}}_1)$ is pt of intersection of tangents drawn at P, Q to ellipse
$\Rightarrow \mathrm{PQ}$ is chord of contact of $\mathrm{R}({\mathrm{x}}_1,{\mathrm{y}}_1)$
$\Rightarrow {\mathrm{xx}}_1+2{\mathrm{yy}}_1-4=0$
This touches circle $\Rightarrow {\mathrm{r}}^2({\mathcal{l}}^2+{\mathrm{m}}^2)={\mathrm{n}}^2$
$\Rightarrow 1({\mathrm{x}}_1^2+4{\mathrm{y}}_1^2)=16$
$\Rightarrow {\mathrm{x}}^2+4{\mathrm{y}}^2=16$ is ellipse

for this ellipse  $\mathrm{e}=\frac{\sqrt{3}}{2};{\mathrm{LL}}^1=2$