A fixed hollow sphere has radius \(a\) and centre \(O\). Points \(A\), \(B\), and \(C\) lie on the inner surface. The horizontal cut through \(B\) and \(C\) is at height \(ka\) above \(O\), where \(0\lt k\lt 1\). The angle between \(OB\) and the upward vertical is \(\theta\). A particle \(P\) of mass \(m\) moves on the smooth inner surface and is projected vertically downwards from \(A\) with speed \(u\).

(a) If \(u=\sqrt{\frac65ga}\) and the reaction at \(B\) is half the reaction at \(A\), find \(k\).

(b) Find \(u\), in terms of \(a\) and \(g\), when the particle just reaches \(B\).

(c) Find \(u\), in terms of \(a\) and \(g\), when the particle passes through \(B\) and subsequently reaches \(C\).