An ellipse passes through the foci of the hyperbola, $9x^2-4y^2=36$  and its major and  minor axes lie along the transverse and conjugate axes of the hyperbola respectively. If  the product of eccentricities of the two conics is  $\frac{1}{2}$, then which of the following points  does not lie on the ellipse?

Equation of hyperbola is  $\frac{x^2}{4}-\frac{y^2}{9}=1$ ; Its focii $=(\pm \sqrt{13},0)$  ;  $e=\frac{\sqrt{13}}{2}$
If e, be the eccentricity of the ellipse, then  $e_1\times \frac{\sqrt{13}}{2}=\frac{1}{2}\Rightarrow e_1=\frac{1}{\sqrt{13}}$
Since ellipse passes through the foci  $k=(\pm \sqrt{13},0)$ of the hyperbola, therefore  $a^2=13$
Now   $\sqrt{a^2-b^2}=a{e}_1$                               $\Rightarrow 13-b^2=1\Rightarrow b^2=12$  
Hence, equation of ellipse is  $\frac{x^2}{13}+\frac{y^2}{12}=1$
Hence the point $(\frac{\sqrt{13}}{2},\frac{\sqrt{3}}{2})$  does not lie on the ellipse.           $(\because S_{11}<0)$