4. For APPLICANTS IN $\left\{ \begin{array} { l } \text { MATHEMATICS } \\ \text { MATHEMATICS \& STATISTICS } \\ \text { MATHEMATICS \& PHILOSOPHY } \end{array} \right\}$ ONLY. Mathematics \& Computer Science and Computer Science applicants should turn to page 14. [Figure] Let $p$ and $q$ be positive real numbers. Let $P$ denote the point ( $p , 0$ ) and $Q$ denote the point $( 0 , q )$. (i) Show that the equation of the circle $C$ which passes through $P , Q$ and the origin $O$ is $$x ^ { 2 } - p x + y ^ { 2 } - q y = 0 .$$ Find the centre and area of $C$. (ii) Show that $$\frac { \text { area of circle } C } { \text { area of triangle } O P Q } \geqslant \pi \text {. }$$ (iii) Find the angles $O P Q$ and $O Q P$ if $$\frac { \text { area of circle } C } { \text { area of triangle } O P Q } = 2 \pi$$
(i) [4 marks] We can complete the squares in $x ^ { 2 } - p x + y ^ { 2 } - q y = 0$ to get
4. For APPLICANTS IN $\left\{ \begin{array} { l } \text { MATHEMATICS } \\ \text { MATHEMATICS \& STATISTICS } \\ \text { MATHEMATICS \& PHILOSOPHY } \end{array} \right\}$ ONLY.\\
Mathematics \& Computer Science and Computer Science applicants should turn to page 14.\\
\includegraphics[max width=\textwidth, alt={}, center]{2b5c0f85-f6ae-4a52-aaf5-c1ed5f1a7e7e-12_687_684_529_680}
Let $p$ and $q$ be positive real numbers. Let $P$ denote the point ( $p , 0$ ) and $Q$ denote the point $( 0 , q )$.\\
(i) Show that the equation of the circle $C$ which passes through $P , Q$ and the origin $O$ is
$$x ^ { 2 } - p x + y ^ { 2 } - q y = 0 .$$
Find the centre and area of $C$.\\
(ii) Show that
$$\frac { \text { area of circle } C } { \text { area of triangle } O P Q } \geqslant \pi \text {. }$$
(iii) Find the angles $O P Q$ and $O Q P$ if
$$\frac { \text { area of circle } C } { \text { area of triangle } O P Q } = 2 \pi$$