Let \(p\) and \(q\) be positive real numbers. Let \(P\) denote the point \((p,0)\) and \(Q\) denote the point \((0,q)\).

  1. Show that the equation of the circle \(C\) which passes through \(P\), \(Q\) and the origin \(O\) is \[x^2 - px + y^2 - qy = 0.\] Find the centre and area of \(C\).

  2. Show that \[\frac{\text{area of circle $C$}}{\text{area of triangle $OPQ$}} \ge \pi.\]

  3. Find the angles \(OPQ\) and \(OQP\) if \[\frac{\text{area of circle $C$}}{\text{area of triangle $OPQ$}} = 2\pi.\]

The areas of the circle and triangle are shown, as is their ratio.