Circumradius of Equilateral Triangle on Lines

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