Let $$A$$ and $$B$$ be points on the two half-lines $$x - \sqrt{3}|y| = \alpha$$, $$\alpha > 0$$, at distance of $$\alpha$$ from the point of intersection $$P$$. The line $$AB$$ meets the angle bisector of the given half-lines at the point $$Q$$. If $$PQ = \frac{9}{2}$$ and $$R$$ is the radius of the circumcircle  of $$\triangle PAB$$, then $$\frac{\alpha^2}{R}$$ is equal to :