The tangent and normal to the ellipse $\frac{{\mathrm{x}}^2}{4}+\frac{{\mathrm{y}}^2}{1}=1$ at the point $\mathrm{P}(2\cos \mathrm{\theta },\sin \mathrm{\theta })$meet the major axis at Q and R, respectively. If QR = 2, then cos $\mathrm{\theta }$ is equal to

$\mathrm{P}(2\cos \mathrm{\theta },\sin \mathrm{\theta })$ be a point 
Equation of tangent at P is $\frac{\mathrm{x}}{2}\cos \mathrm{\theta }+\frac{\mathrm{y}}{1}\sin \mathrm{\theta }=1$
$\Rightarrow$Equation of normal at P is $\frac{2\mathrm{x}}{\cos \mathrm{\theta }}-\frac{\mathrm{y}}{\sin \mathrm{\theta }}=3$
$\begin{array}{l}\therefore \mathrm{Q}=\left(\frac{2}{\cos \mathrm{\theta }},0\right)\mathrm{R}=\left(\frac{3}{2}\cos \mathrm{\theta },0\right)\\ \Rightarrow \mathrm{QR}=2\Rightarrow \left|\frac{2}{\cos \mathrm{\theta }}-\frac{3}{2}\cos \mathrm{\theta }\right|=2\\ \Rightarrow \cos \mathrm{\theta }=\frac{2}{3}(\because \cos \mathrm{\theta }\ne -2)\end{array}$