2020 NYJC CT2 P2 Q3

The diagram below shows a city map of two towns, \(A\) and \(B\) separated by a river. A bridge is to be built between the two towns, which are on opposite sides of a straight river of uniform width \(r\)km, and the two towns are \(p\) km apart measured along the riverbank. Town \(A\) is \(1\)km from the riverbank, and Town \(B\) is \(b\)km away from riverbank.

A bridge is to be built perpendicular to the riverbank at a distance of \(x\)km from Town \(B\), measured along the riverbank, allowing traffic to flow between the two towns.

Find the distance \(x\), in terms of \(b\) and \(p\), such that the distance of travel between Town \(A\) and Town \(B\) can be minimised if \(b>1\). (It is not necessary to verify that the distance is minimum.)