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 14. The line $l$ passes through the origin at angle $2 \alpha$ above the $x$-axis, where $2 \alpha < \frac { \pi } { 2 }$. [Figure] Circles $C _ { 1 }$ of radius 1 and $C _ { 2 }$ of radius 3 are drawn between $l$ and the $x$-axis, just touching both lines. (i) What is the centre of circle $C _ { 1 }$ ? (ii) What is the equation of circle $C _ { 1 }$ ? (iii) For what value of $\alpha$ do circles $C _ { 1 }$ and $C _ { 2 }$ touch? (iv) For this value of $\alpha$ (for which the circles $C _ { 1 }$ and $C _ { 2 }$ touch) a third circle, $C _ { 3 }$, larger than $C _ { 2 }$, is to be drawn between $l$ and the $x$-axis. $C _ { 3 }$ just touches both lines and also touches $C _ { 2 }$. What is the radius of this circle $C _ { 3 }$ ? (v) For the same value of $\alpha$, what is the area of the region bounded by the $x$-axis and the circles $C _ { 1 }$ and $C _ { 2 }$ ? If you require additional space please use the pages at the end of the booklet
(i) [3 marks] Let $d _ { 1 }$ be the distance from $( 0,0 )$ to the point where $C _ { 1 }$ touches the $x$-axis. Note that the $x$-axis is tangent to $C _ { 1 }$ and hence perpendicular to the radius at this point. So $C _ { 1 }$ has centre $\left( d _ { 1 } , 1 \right)$. We have a right-angled triangle, with $\frac { 1 } { d _ { 1 } } = \tan ( \alpha )$, so $d _ { 1 } = \frac { 1 } { \tan ( \alpha ) }$.
\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 14.
The line $l$ passes through the origin at angle $2 \alpha$ above the $x$-axis, where $2 \alpha < \frac { \pi } { 2 }$.\\
\includegraphics[max width=\textwidth, alt={}, center]{a86ae71c-8798-4752-b262-80c21427f83e-14_656_1240_813_376}
Circles $C _ { 1 }$ of radius 1 and $C _ { 2 }$ of radius 3 are drawn between $l$ and the $x$-axis, just touching both lines.\\
(i) What is the centre of circle $C _ { 1 }$ ?\\
(ii) What is the equation of circle $C _ { 1 }$ ?\\
(iii) For what value of $\alpha$ do circles $C _ { 1 }$ and $C _ { 2 }$ touch?\\
(iv) For this value of $\alpha$ (for which the circles $C _ { 1 }$ and $C _ { 2 }$ touch) a third circle, $C _ { 3 }$, larger than $C _ { 2 }$, is to be drawn between $l$ and the $x$-axis. $C _ { 3 }$ just touches both lines and also touches $C _ { 2 }$. What is the radius of this circle $C _ { 3 }$ ?\\
(v) For the same value of $\alpha$, what is the area of the region bounded by the $x$-axis and the circles $C _ { 1 }$ and $C _ { 2 }$ ?
If you require additional space please use the pages at the end of the booklet