9 A smooth hollow hemisphere, of radius \(a\) and centre \(O\), is fixed so that its rim is in a horizontal plane. A smooth uniform \(\operatorname { rod } A B\), of mass \(m\), is in equilibrium, with one end \(A\) resting on the inside of the hemisphere and the point \(C\) on the rod being in contact with the rim of the hemisphere. The rod, of length \(l\), is inclined at an angle \(\theta\) to the horizontal, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{85514b55-3f13-4746-a3ef-747239b64cca-6_453_828_559_591}

1. Explain why the reaction between the rod and the hemisphere at point \(A\) acts through \(O\).
2. Draw a diagram to show the forces acting on the rod.
3. Show that \(l = \frac { 4 a \cos 2 \theta } { \cos \theta }\).