4 A uniform \(\operatorname { rod } A B\) has length \(2 a\) and weight \(W\). The end \(A\) rests on rough horizontal ground and the end \(B\) rests against a smooth vertical wall. The angle between the rod and the horizontal is \(\theta\), where \(\tan \theta = \frac { 4 } { 3 }\). One end of a light inextensible rope is attached to a point \(C\) on the rod. The other end is attached to a point where the vertical wall and the horizontal ground meet. The rope is taut and perpendicular to the rod. The rope and rod are in a vertical plane perpendicular to the wall.

1. Show that \(A C = \frac { 18 } { 25 } a\).
   The magnitude of the frictional force at \(A\) is equal to one quarter of the magnitude of the normal reaction force at \(A\).
2. Show that the tension in the rope is \(\frac { 1 } { 4 } W\).
3. Find expressions, in terms of \(W\), for the magnitudes of the normal reaction forces at \(A\) and \(B\).