The radius of a circle, having minimum area, which touches the curve $y = 4 - x ^ { 2 }$ and the lines $y = | x |$ is:
(1) $2 ( \sqrt { 2 } + 1 )$
(2) $2 ( \sqrt { 2 } - 1 )$
(3) $4 ( \sqrt { 2 } - 1 )$
(4) $4 ( \sqrt { 2 } + 1 )$

If a circle $C$, whose radius is 3 , touches externally the circle $x ^ { 2 } + y ^ { 2 } + 2 x - 4 y - 4 = 0$ at the point $( 2,2 )$, then the length of the intercept cut by this circle $C$ on the $x$-axis is equal to
(1) $2 \sqrt { 3 }$
(2) $\sqrt { 5 }$
(3) $3 \sqrt { 2 }$
(4) $2 \sqrt { 5 }$

If a circle $C$ passing through the point $( 4,0 )$ touches the circle $x ^ { 2 } + y ^ { 2 } + 4 x - 6 y = 12$ externally at the point $( 1 , - 1 )$, then the radius of $C$ is:
(1) 4 units
(2) 5 units
(3) $2 \sqrt { 5 }$ units
(4) $\sqrt { 57 }$ units

The area (in sq. units) of the smaller of the two circles that touch the parabola, $y ^ { 2 } = 4 x$ at the point $( 1,2 )$ and the $x$-axis is
(1) $8 \pi ( 3 - 2 \sqrt { 2 } )$
(2) $8 \pi ( 2 - \sqrt { 2 } )$
(3) $4 \pi ( 3 + \sqrt { 2 } )$
(4) $4 \pi ( 2 - \sqrt { 2 } )$

Two circles each of radius 5 units touch each other at the point $( 1,2 )$. If the equation of their common tangent is $4 x + 3 y = 10$, and $C _ { 1 } ( \alpha , \beta )$ and $C _ { 2 } ( \gamma , \delta ) , C _ { 1 } \neq C _ { 2 }$ are their centres, then $| ( \alpha + \beta ) ( \gamma + \delta ) |$ is equal to

The sum of diameters of the circles that touch (i) the parabola $75x ^ { 2 } = 64(5y - 3)$ at the point $\left(\frac { 8 } { 5 }, \frac { 6 } { 5 }\right)$ and (ii) the $y$-axis, is equal to $\_\_\_\_$.

Two circles in the first quadrant of radii $r _ { 1 }$ and $r _ { 2 }$ touch the coordinate axes. Each of them cuts off an intercept of 2 units with the line $x + y = 2$. Then $r _ { 1 } { } ^ { 2 } + r _ { 2 } { } ^ { 2 } - r _ { 1 } r _ { 2 }$ is equal to $\_\_\_\_$ .

Consider a circle $x - \alpha ^ { 2 } + y - \beta ^ { 2 } = 50$, where $\alpha , \beta > 0$. If the circle touches the line $y + x = 0$ at the point P , whose distance from the origin is $4 \sqrt { 2 }$, then $( \alpha + \beta ) ^ { 2 }$ is equal to $\_\_\_\_$ .

Let the circles $C _ { 1 } : ( x - \alpha ) ^ { 2 } + ( y - \beta ) ^ { 2 } = r _ { 1 } ^ { 2 }$ and $C _ { 2 } : ( x - 8 ) ^ { 2 } + \left( y - \frac { 15 } { 2 } \right) ^ { 2 } = r _ { 2 } ^ { 2 }$ touch each other externally at the point $( 6,6 )$. If the point $( 6,6 )$ divides the line segment joining the centres of the circles $C _ { 1 }$ and $C _ { 2 }$ internally in the ratio $2 : 1$, then $( \alpha + \beta ) + 4 \left( r _ { 1 } ^ { 2 } + r _ { 2 } ^ { 2 } \right)$ equals
(1) 125
(2) 130
(3) 110
(4) 145

A circle $C$ of radius 2 lies in the second quadrant and touches both the coordinate axes. Let $r$ be the radius of a circle that has centre at the point $( 2,5 )$ and intersects the circle $C$ at exactly two points. If the set of all possible values of r is the interval $( \alpha , \beta )$, then $3 \beta - 2 \alpha$ is equal to:
(1) 10
(2) 15
(3) 12
(4) 14

Q66. Let the circles $C _ { 1 } : ( x - \alpha ) ^ { 2 } + ( y - \beta ) ^ { 2 } = r _ { 1 } ^ { 2 }$ and $C _ { 2 } : ( x - 8 ) ^ { 2 } + \left( y - \frac { 15 } { 2 } \right) ^ { 2 } = r _ { 2 } ^ { 2 }$ touch each other externally at the point $( 6,6 )$. If the point $( 6,6 )$ divides the line segment joining the centres of the circles $C _ { 1 }$ and $C _ { 2 }$ internally in the ratio $2 : 1$, then $( \alpha + \beta ) + 4 \left( r _ { 1 } ^ { 2 } + r _ { 2 } ^ { 2 } \right)$ equals
(1) 125
(2) 130
(3) 110
(4) 145

We have a triangle ABC such that
$$\mathrm { AB } = 9 , \quad \mathrm { BC } = 12 , \quad \angle \mathrm { ABC } = 90 ^ { \circ } .$$
There are also two circles $\mathrm { O } _ { 1 }$ and $\mathrm { O } _ { 2 }$ with radii of length $2r$ and $r$, respectively. The two circles $\mathrm { O } _ { 1 }$ and $\mathrm { O } _ { 2 }$ are tangential to each other. Further, $\mathrm { O } _ { 1 }$ is tangent to the two sides AB and AC, and $\mathrm { O } _ { 2 }$ is tangent to the two sides CA and CB. We are to find the value of $r$.
First, let D and E denote the points at which the segment AC is tangent to the circles $\mathrm { O } _ { 1 }$ and $\mathrm { O } _ { 2 }$ respectively, and set $\alpha = \angle \mathrm { O } _ { 1 } \mathrm { AC }$. Then, since $\tan 2 \alpha = \frac { \square \mathbf { A } } { \square }$, we have $\tan \alpha = \frac { \mathbf { C } } { \mathbf { D } }$ using the double-angle formula. Thus, we obtain $\mathrm { AD } = \mathbf { E }$.
Next, set $\beta = \angle \mathrm { O } _ { 2 } \mathrm { CA }$. Since $\alpha + \beta = \mathbf { F G } ^ { \circ }$, we have $\tan \beta = \frac { \mathbf { H } } { \square \mathbf{I} }$ using the addition theorem. Thus, we obtain $\mathrm { CE } = \square r$.
Moreover, it follows that $\mathrm { AC } = \mathbf { K L }$ and $\mathrm { DE } = \mathbf { M } \sqrt { \mathbf { N } } r$.
Finally we obtain
$$r = \frac { \mathbf { O P } ( \mathbf { Q } - \mathbf { R } \sqrt { \mathbf { S } } ) } { 41 }$$

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

On a coordinate plane, there is a circle with center $A(a, b)$ that is tangent to both coordinate axes. There is also a point $P(c, c)$, where $a > c > 0$, and it is known that $\overline{PA} = a + c$. Select the correct options.
(1) $a = b$ (2) Point $P$ is on the line $x + y = 0$ (3) Point $P$ is inside the circle (4) $\frac{a + c}{b - c} = \sqrt{2}$ (5) $\frac{a}{c} = 2 + 3\sqrt{2}$

Find the shortest distance between the two circles with equations:
$$\begin{aligned} & ( x + 2 ) ^ { 2 } + ( y - 3 ) ^ { 2 } = 18 \\ & ( x - 7 ) ^ { 2 } + ( y + 6 ) ^ { 2 } = 2 \end{aligned}$$
A 0
B 4
C 16
D $2 \sqrt { 2 }$
E $5 \sqrt { 2 }$

The circles with equations
$$\begin{aligned} &(x+4)^2 + (y+1)^2 = 64 \quad \text{and} \\ &(x-8)^2 + (y-4)^2 = r^2 \quad \text{where } r > 0 \end{aligned}$$
have exactly one point in common. Find the difference between the two possible values of $r$.

A circle has equation
$$x ^ { 2 } + a x + y ^ { 2 } + b y + c = 0$$
where $a , b$ and $c$ are non-zero real constants. Which one of the following is a necessary and sufficient condition for the circle to be tangent to the $y$-axis?
A $a ^ { 2 } = 4 c$
B $b ^ { 2 } = 4 c$
C $\frac { a } { 2 } = \sqrt { \frac { a ^ { 2 } + b ^ { 2 } } { 4 } - c }$
D $\frac { b } { 2 } = \sqrt { \frac { a ^ { 2 } + b ^ { 2 } } { 4 } - c }$ $\mathbf { E } \quad - \frac { a } { 2 } = \sqrt { \frac { a ^ { 2 } + b ^ { 2 } } { 4 } - c }$ $\mathbf { F } \quad - \frac { b } { 2 } = \sqrt { \frac { a ^ { 2 } + b ^ { 2 } } { 4 } - c }$

Below, two concentric circles with center $O$ and a circle with center $M$ tangent to both circles are given.
The radius of the small circle with center $O$ is 4 units less than the radius of the large circle with center $O$, and 1 unit more than the radius of the circle with center $M$.
Accordingly, what is the area of the shaded region in square units?
A) $28 \pi$ B) $32 \pi$ C) $36 \pi$ D) $39 \pi$ E) $45 \pi$