ACJC Graphing techniques Tutorial Q5 (*)

Show circle \(C\) passing through \(P\left( p,0 \right)\), \(Q\left( 0,q \right)\)and the origin has equation \({{x}^{2}}-px+{{y}^{2}}-qy=0\) (*)

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

1. Show that the equation of the circle \(C\) which passes through \(P\), \(Q\) and the origin \(O\) is \({{x}^{2}}-px+{{y}^{2}}-qy=0\).
2. Find the area of \(C\) in terms of \(p\) and \(q\), and show that \(\frac{\mathrm{area}\,\mathrm{of}\,\mathrm{circle}\,\,\,C}{\mathrm{area}\,\mathrm{of}\,\mathrm{triangle}\,\,\,OPQ}\ge \pi \).
3. Find the angle \(OPQ\) if \(\frac{\mathrm{area}\,\mathrm{of}\,\mathrm{circle}\,\,\,C}{\mathrm{area}\,\mathrm{of}\,\mathrm{triangle}\,\,\,OPQ}=2\pi \) and \(p>q\).