2021 ACJC P1 Q5

[It is given that a cone of radius \(r\) and height \(h\) has volume \(\frac{1}{3}\pi {{r}^{2}}h\).]

A sphere with fixed radius \(a\) and centre \(O\) is inscribed in a right cone with base radius \(r\) and height \(h\). The sphere is in contact with the base and the curved surface of the cone, as shown in the diagram below.

1. Show that \({{r}^{2}}=\frac{{{a}^{2}}h}{h-2a}\).[2]
2. Use differentiation to find, in terms of \(a\), the minimum volume of the cone, proving that it is a minimum.[6]