If the tangent & normal to the ellipse ${\mathrm{x}}^2+4{\mathrm{y}}^2=4$ $\text{ at a point }\mathrm{p}\left(\mathrm{\theta }\right)\text{ on it meet the major axis in }\mathrm{Q}$ & R respectively if $0<\mathrm{\theta }<\frac{\mathrm{\pi }}{2}\mathrm{\&}\mathrm{QR}=2\text{ then }$$\text{ S.T }\mathrm{\theta }={\cos }^{-1}\left(\frac{2}{3}\right)$ 

Equation of the ellipse is ${\mathrm{x}}^2+4{\mathrm{y}}^2=4\Rightarrow \frac{{\mathrm{x}}^2}{4}+{\mathrm{y}}^2=1$
Let $\mathrm{\theta }$ be the eccentric angle of P.
$\text{ Then }\mathrm{P}=(2\cos \mathrm{\theta },\sin \mathrm{\theta })\text{. }$
Equation of the tangent at P is $\left(\frac{\cos \mathrm{\theta }}{2}\right)\mathrm{x}+(\sin \mathrm{\theta })\mathrm{y}=1$
Tangent at P meets major axis y = 0 at $\mathrm{Q}=\left(\frac{2}{\cos \mathrm{\theta }},0\right)$
Equation of the normal at P is $\frac{2\mathrm{x}}{\cos \mathrm{\theta }}-\frac{\mathrm{y}}{\sin \mathrm{\theta }}=4-1\Rightarrow \left(\frac{2}{\cos \mathrm{\theta }}\right)\mathrm{x}-\frac{\mathrm{y}}{\sin \mathrm{\theta }}=3$
Normal at P meets major axis y = 0 at R
$\begin{array}{l}\Rightarrow \mathrm{R}=\left(\frac{3\cos \mathrm{\theta }}{2},0\right)\\ Now \mathrm{QR}=2\Rightarrow {\left(\frac{3\cos \mathrm{\theta }}{2}-\frac{2}{\cos \mathrm{\theta }}\right)}^2=4\\ \Rightarrow \frac{9{\cos }^2\mathrm{\theta }}{4}+\frac{4}{{\cos }^2\mathrm{\theta }}-6=4\\ \Rightarrow 9{\cos }^4\mathrm{\theta }-40{\cos }^2\mathrm{\theta }+16=0\\ \Rightarrow \left(9{\cos }^2\mathrm{\theta }-4\right)\left({\cos }^2\mathrm{\theta }-4\right)=0\\ \Rightarrow 9{\cos }^2\mathrm{\theta }-4=0\Rightarrow \cos \mathrm{\theta }=\pm \frac{2}{3}\\ \Rightarrow \mathrm{\theta }={\cos }^{-1}\left(\frac{2}{3}\right)\text{ or }\mathrm{\pi }-{\cos }^{-1}\left(\frac{2}{3}\right)\end{array}$