\section*{4. For APPLICANTS IN $\left\{ \begin{array} { l } \text { MATHEMATICS } \\ \text { MATHEMATICS \& STATISTICS } \\ \text { MATHEMATICS \& PHILOSOPHY } \end{array} \right\}$ ONLY.}
Mathematics \& Computer Science, Computer Science and Computer Science \& Philosophy applicants should turn to page 16.

A circle $A$ passes through the points $( - 1,0 )$ and $( 1,0 )$. Circle $A$ has centre $( m , h )$, and radius $r$.\\
(i) Determine $m$ and write $r$ in terms of $h$.\\
(ii) Given a third point $\left( x _ { 0 } , y _ { 0 } \right)$ and $y _ { 0 } \neq 0$ show that there is a unique circle passing through the three points $( - 1,0 ) , ( 1,0 ) , \left( x _ { 0 } , y _ { 0 } \right)$.

For the remainder of the question we consider three circles $A , B$, and $C$, each passing through the points $( - 1,0 ) , ( 1,0 )$. Each circle is cut into regions by the other two circles. For a group of three such circles, we will say the lopsidedness of a circle is the fraction of the full area of that circle taken by its largest region.\\
(iii) Let circle $A$ additionally pass through the point ( 1,2 ), circle $B$ pass through ( 0,1 ), and let circle $C$ pass through the point $( 0 , - 4 )$. What is the lopsidedness of circle $A$ ?\\
(iv) Let $p > 0$. Now let $A$ pass through ( $1,2 p$ ), $B$ pass through ( 0,1 ), and $C$ pass through $( - 1 , - 2 p )$. Show that the value of $p$ minimising the lopsidedness of circle $B$ satisfies the equation

$$\left( p ^ { 2 } + 1 \right) \tan ^ { - 1 } \left( \frac { 1 } { p } \right) - p = \frac { \pi } { 6 }$$

Note that $\tan ^ { - 1 } ( x )$ is sometimes written as $\arctan ( x )$ and is the value of $\theta$ in the range $\frac { - \pi } { 2 } < \theta < \frac { \pi } { 2 }$ such that $\tan ( \theta ) = x$.


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