Two parabolas have the same focus $(4,3)$ and their directrices are the $x$-axis and the $y$-axis, respectively. If these parabolas intersect at the points $A$ and $B$, then $(AB)^2$ is equal to:
(1) 392
(2) 384
(3) 192
(4) 96

Q84. Consider the circle $C : x ^ { 2 } + y ^ { 2 } = 4$ and the parabola $P : y ^ { 2 } = 8 x$. If the set of all values of $\alpha$, for which three chords of the circle $C$ on three distinct lines passing through the point $( \alpha , 0 )$ are bisected by the parabola $P$ is the interval $( p , q )$, then $( 2 q - p ) ^ { 2 }$ is equal to $\_\_\_\_$

3. Let $O , P , P _ { 1 } , P _ { 2 }$ be the points in the $( x , y )$-plane with coordinates $( 0,0 ) , ( s , 1 / s )$, $\left( s _ { 1 } , 1 / s _ { 1 } \right) , \left( s _ { 2 } , 1 / s _ { 2 } \right)$ respectively.
(i) Using the axes below, sketch the curve traced out by $P$ as $s$ varies over non-zero real values, and find an equation for the curve in the form $y = f ( x )$.
(ii) Write down the equation of the straight line $P P _ { 1 }$ joining $P$ to $P _ { 1 }$, giving your answer in the form $y = m _ { 1 } x + c _ { 1 }$.
(iii) Show that the line $P P _ { 1 }$ is perpendicular to $P P _ { 2 }$ if, and only if, $s _ { 1 } s _ { 2 } = - 1 / s ^ { 2 }$.
(iv) Let $m _ { 1 } , m _ { 2 } , n _ { 1 } , n _ { 2 }$ be the gradients of the lines $P P _ { 1 } , P P _ { 2 } , O P _ { 1 } , O P _ { 2 }$ respectively. Show that
$$\left( \frac { m _ { 1 } } { m _ { 2 } } \right) ^ { 2 } = \frac { n _ { 1 } } { n _ { 2 } }$$
[Figure]

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.

3. For APPLICANTS IN $\left\{ \begin{array} { l } \text { MATHEMATICS } \\ \text { MATHEMATICS \& STATISTICS } \\ \text { MATHEMATICS \& PHILOSOPHY } \\ \text { MATHEMATICS \& COMPUTER SCIENCE } \end{array} \right\}$ ONLY.

Computer Science and Computer Science \& Philosophy applicants should turn to page 20.
Below is a sketch of the curve $S$ with equation $y ^ { 2 } - y = x ^ { 3 } - x$. The curve crosses the $x$-axis at the origin and at $( a , 0 )$ and at $( b , 0 )$ for some real numbers $a < 0$ and $b > 0$. The curve only exists for $\alpha \leqslant x \leqslant \beta$ and for $x \geqslant \gamma$. The three points with coordinates $( \alpha , \delta ) , ( \beta , \delta )$, and $( \gamma , \delta )$ are all on the curve. [Figure]
(i) What are the values of $a$ and $b$ ?
(ii) By completing the square, or otherwise, find the value of $\delta$.
(iii) Explain why the curve is symmetric about the line $y = \delta$.
(iv) Find a cubic equation in $x$ which has roots $\alpha , \beta , \gamma$. (Your expression for the cubic should not involve $\alpha , \beta$, or $\gamma$ ). Justify your answer.
(v) By considering the factorization of this cubic, find the value of $\alpha + \beta + \gamma$.
(vi) Let $C$ denote the circle which has the points $( \alpha , \delta )$ and $( \beta , \delta )$ as ends of a diameter. Write down the equation of $C$. Show that $C$ intersects $S$ at two other points and find their common $x$-co-ordinate in terms of $\gamma$.
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4. On the coordinate plane, how many intersection points do the graphs of the equation $\frac{x^2}{9} + \frac{y^2}{4} = 1$ and $\frac{(x+1)^2}{16} - \frac{y^2}{9} = 1$ have?
(1) 1
(2) 2
(3) 3
(4) 4
(5) 0

On the coordinate plane, there is a regular hexagon $A B C D E F$ with side length 3, where $A ( 3,0 ) , D ( - 3,0 )$. How many intersection points does the ellipse $\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 7 } = 1$ have with the regular hexagon $A B C D E F$?
(1) 0
(2) 2
(3) 4
(4) 6
(5) 8

Two circles have the same radius. The centre of one circle is $( - 2,1 )$. The centre of the other circle is $( 3 , - 2 )$. The circles intersect at two distinct points. What is the equation of the straight line through the two points at which the circles intersect?
A $3 x - 5 y = 4$ B $3 x + 5 y = - 1$ C $5 x - 3 y = - 4$ D $5 x - 3 y = - 1$ E $\quad 5 x - 3 y = 1$ F $5 x - 3 y = 4$ G $5 x + 3 y = 1$

$$|\mathrm{OM}| = 2 \text{ units}$$
In the rectangular coordinate plane, a semicircle with center at point M and a quarter circle with center at the origin intersect at point A as shown in the figure.
Accordingly, what is the x-coordinate of point A?
A) $\frac{5}{3}$ B) $\sqrt{2}$ C) $\frac{\sqrt{3}}{2}$ D) $\frac{3}{2}$ E) $\sqrt{3}$