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.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{f229588a-5602-44a5-acb0-6efec65f41af-12_640_643_653_310}
\captionsetup{labelformat=empty}
\caption{Diagram when $h > 2 / \sqrt { 5 }$}
\end{center}
\end{figure}

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{f229588a-5602-44a5-acb0-6efec65f41af-12_636_641_657_1098}
\captionsetup{labelformat=empty}
\caption{Diagram when $h < \sqrt { 3 } / 2$}
\end{center}
\end{figure}

The three corners of a triangle $T$ are $( 0,0 ) , ( 3,0 ) , ( 1,2 h )$ where $h > 0$. The circle $C$ has equation $x ^ { 2 } + y ^ { 2 } = 4$. The angle of the triangle at the origin is denoted as $\theta$. The circle and triangle are drawn in the diagrams above for different values of $h$.\\
(i) Express $\tan \theta$ in terms of $h$.\\
(ii) Show that the point $( 1,2 h )$ lies inside $C$ when $h < \sqrt { 3 } / 2$.\\
(iii) Find the equation of the line connecting $( 3,0 )$ and $( 1,2 h )$. Show that this line is tangential to the circle $C$ when $h = 2 / \sqrt { 5 }$.\\
(iv) Suppose now that $h > 2 / \sqrt { 5 }$. Find the area of the region inside both $C$ and $T$ in terms of $\theta$.\\
(v) Now let $h = 6 / 7$. Show that the point ( $8 / 5,6 / 5$ ) lies on both the line (from part (iii)) and the circle $C$.

Hence show that the area of the region inside both $C$ and $T$ equals

$$\frac { 27 } { 35 } + 2 \alpha$$

where $\alpha$ is an angle whose tangent, $\tan \alpha$, you should determine.\\[0pt]
[You may use the fact that the area of a triangle with corners $( 0,0 ) , ( a , b ) , ( c , d )$ equals $\frac { 1 } { 2 } | a d - b c |$.]

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