A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end is attached to a fixed point \(O\). The particle is held with the string taut and making an angle \(\theta\) with the downward vertical through \(O\).

The particle is projected at right angles to the string with speed \(\dfrac13\sqrt{10ag}\) and begins to move downwards along a circular path. When the string is vertical, it strikes a small smooth peg at point \(A\), vertically below \(O\). The circle changes to have centre \(A\). When the string makes an angle \(\theta\) with the upward vertical through \(A\), the string becomes slack. The distance of \(A\) below \(O\) is \(\dfrac59a\).

(a) Find \(\cos\theta\).

(b) Find the ratio of the tensions in the string immediately before and immediately after it strikes the peg.