The smooth vertical walls \(A B\) and \(C B\) are at right angles to each other. A particle \(P\) is moving with speed \(u\) on a smooth horizontal floor and strikes the wall \(C B\) at an angle \(\alpha\). It rebounds at an angle \(\beta\) to the wall \(C B\). The particle then strikes the wall \(A B\) and rebounds at an angle \(\gamma\) to that wall (see diagram). The coefficient of restitution between each wall and \(P\) is \(e\).
(a) Show that \(\tan \beta=e \tan \alpha\).

(b) Express \(\gamma\) in terms of \(\alpha\) and explain what this result means about the final direction of motion of \(P\).

As a result of the two impacts the particle loses \(\frac{8}{9}\) of its initial kinetic energy.
(c) Given that \(\alpha+\beta=90^{\circ}\), find the value of \(e\) and the value of \(\tan \alpha\).