The Focii of the Ellipse $\frac{{\mathrm{x}}^2}{16}+\frac{{\mathrm{y}}^2}{12}=1$ are S and S¹. The normal at P meets the major axis at Q. Then $\frac{{\mathrm{PQ}}^2}{\mathrm{SP}\cdot {\mathrm{S}}^1\mathrm{P}}$ is

$\begin{array}{l}Given ellipse \frac{x^2}{16}+\frac{y^2}{12}=1\\ \mathrm{where} \mathrm{a}=4,\mathrm{b}=2\sqrt{3},\mathrm{e}=\frac{1}{2}\\ foci are (\pm 2,0)\\ \Rightarrow S=(2,0) and S\text{'}=(-2,0)\\ \Rightarrow \mathrm{P}=(4\cos \mathrm{\theta },2\sqrt{3}\sin \mathrm{\theta })\mathrm{Q}(\cos \mathrm{\theta },0)\\ Equation of normal at P is \\ \frac{4x}{\cos \theta }-\frac{2\sqrt{3}y}{\sin \theta }=4\\ This normal meets major axis at Q=(\cos \theta ,0)\\ \Rightarrow \frac{(\mathrm{PQ}{)}^2}{\mathrm{Sp}\cdot {\mathrm{S}}^{\mathrm{\prime }}\mathrm{P}}=\frac{9+3{\sin }^2 \mathrm{\theta }}{4\left(3+{\sin }^2 \mathrm{\theta }\right)}=\frac{3}{4}\end{array}$