The angle between the line joining the origin to the points of intersection of the line $\sqrt{3}x+y=2$ and the curve y² - x² =4 is

 Homogenizing the hyperbola using the straight line, we get a pair of straight lines OP and OQ, which are given by
$\begin{array}{l}{\mathrm{y}}^2-{\mathrm{x}}^2=4{\left(\frac{\sqrt{3}\mathrm{x}+\mathrm{y}}{2}\right)}^2\\ \mathrm{or} {\mathrm{y}}^2-{\mathrm{x}}^2=3{\mathrm{x}}^2+{\mathrm{y}}^2+2\sqrt{3}\mathrm{xy}\\ \mathrm{or} 4{\mathrm{x}}^2+2\sqrt{3}\mathrm{xy}=0\\ \mathrm{or} \mathrm{x}=0\text{ and }2\mathrm{x}+\sqrt{3}\mathrm{y}=0\end{array}$

Angle between the lines is 

$\frac{\pi }{2}-{\tan }^{-1}\left(\frac{2}{\sqrt{3}}\right)={\tan }^{-1}\left(\frac{\sqrt{3}}{2}\right)$