A particle \(P\) is projected with speed \(u\) at an angle \(\alpha\) above the horizontal from a point \(O\) on a horizontal plane and moves freely under gravity. The horizontal and vertical displacements of \(P\) from \(O\) at a subsequent time \(t\) are denoted by \(x\) and \(y\) respectively.
(a) Derive the equation of the trajectory of \(P\) in the form
\(y=x \tan \alpha-\frac{g x^{2}}{2 u^{2}} \sec ^{2} \alpha\)

The point \(Q\) is the highest point on the trajectory of \(P\) in the case where \(\alpha=45^{\circ}\).
(b) Show that the \(x\)-coordinate of \(Q\) is \(\frac{u^{2}}{2 g}\).
(c) Find the other value of \(\alpha\) for which \(P\) would pass through the point \(Q\).