IP1U13 SWII (Pythagoras) Q24

The distance between a pair of parallel lines, \(a\) and \(b\), is \(4\) cm. The distance from point \(A\) to line \(a\) is \(2\) cm, The distance from point \(B\) to line \(b\) is \(3\) cm, and \(AB=2\sqrt{30}\).

Find a point \(M\) on line \(a\) and a point \(N\) on line \(b\) such that \(MN \perp a\) and the length of \(AM + MN + NB\) is the shortest. Hence find the shortest length of \(AM + NB\)