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.
In this question we will consider subsets $S$ of the $x y$-plane and points $( a , b )$ which may or may not be in $S$. We will be interested in those points of $S$ which are nearest to the point $( a , b )$. There may be many such points, a unique such point, or no such point.
(i) Let $S$ be the disc $x ^ { 2 } + y ^ { 2 } \leqslant 1$. For a given point ( $a , b$ ), find the unique point of $S$ which is closest to $( a , b )$. [0pt] [You will need to consider separately the cases when $a ^ { 2 } + b ^ { 2 } > 1$ and when $a ^ { 2 } + b ^ { 2 } \leqslant 1$.]
(ii) Describe (without further justification) an example of a subset $S$ and a point $( a , b )$ such that there is no point of $S$ nearest to $( a , b )$.
(iii) Describe (without further justification) an example of a subset $S$ and a point $( a , b )$ such that there is more that one point of $S$ nearest to $( a , b )$.
(iv) Let $S$ denote the line with equation $y = m x + c$. Obtain an expression for the distance of $( a , b )$ from a general point $( x , m x + c )$ of $S$.
Show that there is a unique point of $S$ nearest to $( a , b )$.
(v) For some subset $S$, and for any point ( $a , b$ ), the nearest point of $S$ to ( $a , b$ ) is
$$\left( \frac { a + 2 b - 2 } { 5 } , \frac { 2 a + 4 b + 1 } { 5 } \right) .$$
Describe the subset $S$.
(vi) Say now that $S$ has the property that for any two points $P$ and $Q$ in $S$ the line segment $P Q$ is also in $S$.
Show that, for a given point $( a , b )$, there cannot be two distinct points of $S$ which are nearest to $( a , b )$.
If you require additional space please use the pages at the end of the booklet

On a coordinate plane, there is a circle with radius 7 and center $O$. It is known that points $A, B$ are on the circle and $\overline{AB} = 8$. Then the dot product $\overrightarrow{OA} \cdot \overrightarrow{OB} =$ (14--1) (14--2).

A circle has equation $( x - 9 ) ^ { 2 } + ( y + 2 ) ^ { 2 } = 4$
A square has vertices at $( 1,0 ) , ( 1,2 ) , ( - 1,2 )$ and $( - 1,0 )$.
A straight line bisects both the area of the circle and the area of the square. What is the $x$-coordinate of the point where this straight line meets the $x$-axis?
A 2
B 3
C 4
D 4.5
E 5
F 6
G The straight line is not uniquely determined by the information given, so there is more than one possible point of intersection. $\mathbf { H }$ There is no straight line that bisects both the area of the circle and the area of the square.

Point $P$ lies on the circle with equation $( x - 2 ) ^ { 2 } + ( y - 1 ) ^ { 2 } = 16$
Point $Q$ lies on the circle with equation $( x - 4 ) ^ { 2 } + ( y + 5 ) ^ { 2 } = 16$
What is the maximum possible length of $P Q$ ?

In the Cartesian coordinate plane, two circles with one centered at $(12,0)$ and the other centered at $(0,9)$ intersect only at point $(4,6)$.
What is the distance between the points on these circles that are closest to the origin?
A) $\sqrt { 5 }$ B) $\sqrt { 10 }$ C) $\sqrt { 13 }$ D) $2 \sqrt { 5 }$ E) $2 \sqrt { 10 }$