Let \( A, B \) be points on the two half-lines \( x - \sqrt{3}\,|y| = \alpha, \; \alpha > 0 \) at a distance of \( \alpha \) 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 \( \triangle PAB \), then \( \dfrac{\alpha^{2}}{R} \) is equal to ______.
