Two uniform smooth spheres \(A\) and \(B\) of equal radii have masses \(m\) and \(k m\) respectively. The two spheres are on a horizontal surface. Sphere \(A\) is travelling with speed \(u\) towards sphere \(B\) which is at rest. The spheres collide. Immediately before the collision, the direction of motion of \(A\) makes an angle \(\alpha\) with the line of centres. The coefficient of restitution between the spheres is \(\frac{1}{2}\).
(a) Show that the speed of \(B\) after the collision is \(\frac{3 u \cos \alpha}{2(1+k)}\) and find also an expression for the speed of \(A\) along the line of centres after the collision, in terms of \(k, u\) and \(\alpha\).

After the collision, the kinetic energy of \(A\) is equal to the kinetic energy of \(B\).
(b) Given that \(\tan \alpha=\frac{2}{3}\), find the possible values of \(k\).