Show circle $C$ passing through $P\left( p,0 \right)$, $Q\left( 0,q \right)$and the origin has equat

Question

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 $\left( 0,q \right)$. 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 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 $. Find the angle $OPQ$ if $\frac{\mathrm{area}\,\mathrm{of}\,\mathrm{circle}\,\,\,C}{\mathrm{area}\,\mathrm{of}\,\mathrm{triangle}\,\,\,OPQ}=2\pi $ and $p>q$.

Accepted Answer

(i) $x^2 - px + y^2 - qy = 0$ (shown) (ii) Ratio $\geq \pi$ (shown) (iii) $\angle OPQ = 15°$

ACJC Graphing techniques Tutorial Q5 (*)

ACJC Graphing techniques Tutorial Q5 (*)

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