The ratio of the area enclosed by the locus of mid-point of PS and area of the ellipse where P is any point on the ellipse and S is the focus of the ellipse, is

Ellipse equation is $\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}+\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}=1,\mathrm{Area}=\mathrm{\pi ab}$
Let $\mathrm{P}=(\mathrm{acos\theta },\mathrm{bsin\theta })$
$\mathrm{S}=(\mathrm{ae},0)$
$\mathrm{M}(\mathrm{h},\mathrm{k})$ mid point of PS
$\Rightarrow \mathrm{h}=\frac{\mathrm{ae}+\mathrm{acos\theta }}{2};\mathrm{k}=\frac{\mathrm{bsin\theta }}{2}$
$={\left(\frac{\mathrm{h}-\frac{\mathrm{ae}}{2}}{\mathrm{a}/2}\right)}^2+\frac{{\mathrm{k}}^2}{({\mathrm{b}}^2/4)}=1$, locus of (h,k) is ellipse
$\mathrm{Area}=\mathrm{\pi }\left(\frac{{\displaystyle \mathrm{a}}}{{\displaystyle 2}}\right)\left(\frac{{\displaystyle \mathrm{b}}}{{\displaystyle 2}}\right)=\frac{1}{4}\mathrm{\pi ab}$