5 A uniform ladder \(A B\), of weight \(W\) and length \(2 a\), rests with the end \(A\) in contact with rough horizontal ground and the end \(B\) resting against a smooth vertical wall. The ladder is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 12 } { 13 }\). A man of weight 6 W is standing on the ladder at a distance \(x\) from \(A\) and the system is in equilibrium.

1. Show that the magnitude of the frictional force exerted by the ground on the ladder is \(\frac { 5 W } { 24 } \left( 1 + \frac { 6 x } { a } \right)\). The coefficient of friction between the ladder and the ground is \(\frac { 1 } { 3 }\).
2. Find, in terms of \(a\), the greatest value of \(x\) for which the system is in equilibrium. The bottom of the ladder \(A\) is moved closer to the wall so that the ladder is now inclined at an angle \(\alpha\) to the horizontal. The man of weight 6 W can now stand at the top of the ladder \(B\) without the ladder slipping.
3. Find the least possible value of \(\tan \alpha\).