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 diagram below shows the parabola $y = x ^ { 2 }$ and a circle with centre $( 0,2 )$ just 'resting' on the parabola. By 'resting' we mean that the circle and parabola are tangential to each other at the points $A$ and $B$. [Figure] (i) Let ( $x , y$ ) be a point on the parabola such that $x \neq 0$. Show that the gradient of the line joining this point to the centre of the circle is given by $$\frac { x ^ { 2 } - 2 } { x } .$$ (ii) With the help of the result from part (i), or otherwise, show that the coordinates of $B$ are given by $$\left( \sqrt { \frac { 3 } { 2 } } , \frac { 3 } { 2 } \right) .$$ (iii) Show that the area of the sector of the circle enclosed by the radius to $A$, the minor $\operatorname { arc } A B$ and the radius to $B$ is equal to $$\frac { 7 } { 4 } \cos ^ { - 1 } \left( \frac { 1 } { \sqrt { 7 } } \right)$$ (iv) Suppose now that a circle with centre ( $0 , a$ ) is resting on the parabola, where $a > 0$. Find the range of values of $a$ for which the circle and parabola touch at two distinct points. (v) Let $r$ be the radius of a circle with centre ( $0 , a$ ) that is resting on the parabola. Express $a$ as a function of $r$, distinguishing between the cases in which the circle is, and is not, in contact with the vertex of the parabola.
(i) [2 marks] We need the gradient of the line segment joining ( $x , x ^ { 2 }$ ) to $C = ( 0,2 )$. This is, for $x \ne
\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 diagram below shows the parabola $y = x ^ { 2 }$ and a circle with centre $( 0,2 )$ just 'resting' on the parabola. By 'resting' we mean that the circle and parabola are tangential to each other at the points $A$ and $B$.\\
\includegraphics[max width=\textwidth, alt={}, center]{57150e89-9c08-4eab-97c4-06ccb6339e05-12_545_519_820_764}\\
(i) Let ( $x , y$ ) be a point on the parabola such that $x \neq 0$. Show that the gradient of the line joining this point to the centre of the circle is given by
$$\frac { x ^ { 2 } - 2 } { x } .$$
(ii) With the help of the result from part (i), or otherwise, show that the coordinates of $B$ are given by
$$\left( \sqrt { \frac { 3 } { 2 } } , \frac { 3 } { 2 } \right) .$$
(iii) Show that the area of the sector of the circle enclosed by the radius to $A$, the minor $\operatorname { arc } A B$ and the radius to $B$ is equal to
$$\frac { 7 } { 4 } \cos ^ { - 1 } \left( \frac { 1 } { \sqrt { 7 } } \right)$$
(iv) Suppose now that a circle with centre ( $0 , a$ ) is resting on the parabola, where $a > 0$. Find the range of values of $a$ for which the circle and parabola touch at two distinct points.\\
(v) Let $r$ be the radius of a circle with centre ( $0 , a$ ) that is resting on the parabola. Express $a$ as a function of $r$, distinguishing between the cases in which the circle is, and is not, in contact with the vertex of the parabola.