The points P, Q and the origin forming a right-angled triangle as will be described. A circle is marked around the triangle touching all three vertices.

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