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