Two uniform smooth spheres \(A\) and \(B\) of equal radii each have mass \(m\). The two spheres are each moving with speed \(u\) on a horizontal surface when they collide. Immediately before the collision, \(A\) 's direction of motion makes an angle \(\alpha\) with the line of centres, and \(B\) 's direction of motion makes an angle \(\beta\) with the line of centres (see diagram). The coefficient of restitution between the spheres is \(\frac{1}{3}\) and \(2 \cos \beta=\cos \alpha\).
(a) Show that the direction of motion of \(A\) after the collision is perpendicular to the line of centres.

The total kinetic energy of the spheres after the collision is \(\frac{3}{4} m u^{2}\).
(b) Find the value of \(\alpha\).