If the length of the latus rectum of a parabola, whose focus is $( a , a )$ and the tangent at its vertex is $x + y = a$, is 16 , then $| a |$ is equal to
(1) $2 \sqrt { 2 }$
(2) $2 \sqrt { 3 }$
(3) $4 \sqrt { 2 }$
(4) 4

Let the centre of a circle $C$ be $( \alpha , \beta )$ and its radius $r < 8$. Let $3 x + 4 y = 24$ and $3 x - 4 y = 32$ be two tangents and $4 x + 3 y = 1$ be a normal to $C$. Then $( \alpha - \beta + r )$ is equal to
(1) 7
(2) 5
(3) 6
(4) 9

The points of intersection of the line $a x + b y = 0 , ( \mathrm { a } \neq \mathrm { b } )$ and the circle $\mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } - 2 \mathrm { x } = 0$ are $A ( \alpha , 0 )$ and $B ( 1 , \beta )$. The image of the circle with $A B$ as a diameter in the line $\mathrm { x } + \mathrm { y } + 2 = 0$ is:
(1) $x ^ { 2 } + y ^ { 2 } + 5 x + 5 y + 12 = 0$
(2) $x ^ { 2 } + y ^ { 2 } + 3 x + 5 y + 8 = 0$
(3) $x ^ { 2 } + y ^ { 2 } + 3 x + 3 y + 4 = 0$
(4) $x ^ { 2 } + y ^ { 2 } - 5 x - 5 y + 12 = 0$

Consider a circle $C _ { 1 } : x ^ { 2 } + y ^ { 2 } - 4 x - 2 y = \alpha - 5$. Let its mirror image in the line $y = 2 x + 1$ be another circle $C _ { 2 } : 5 x ^ { 2 } + 5 y ^ { 2 } - 10 f x - 10 g y + 36 = 0$. Let $r$ be the radius of $C _ { 2 }$. Then $\alpha + r$ is equal to $\_\_\_\_$

If one of the diameters of the circle $x ^ { 2 } + y ^ { 2 } - 10 x + 4 y + 13 = 0$ is a chord of another circle $C$, whose center is the point of intersection of the lines $2 x + 3 y = 12$ and $3 x - 2 y = 5$, then the radius of the circle $C$ is
(1) $\sqrt { 20 }$
(2) 4
(3) 6
(4) $3 \sqrt { 2 }$

Let $C$ be the circle of minimum area touching the parabola $y = 6 - x ^ { 2 }$ and the lines $y = \sqrt { 3 } | x |$. Then, which one of the following points lies on the circle $C$ ?
(1) $( 1,2 )$
(2) $( 1,1 )$
(3) $( 2,2 )$
(4) $( 2,4 )$

Four distinct points $( 2 \mathrm { k } , 3 \mathrm { k } ) , ( 1,0 ) , ( 0,1 )$ and $( 0,0 )$ lie on a circle for $k$ equal to:
(1) $\frac { 2 } { 13 }$
(2) $\frac { 3 } { 13 }$
(3) $\frac { 5 } { 13 }$
(4) $\frac { 1 } { 13 }$

Let $A(\alpha, 0)$ and $B(0, \beta)$ be the points on the line $5x + 7y = 50$. Let the point $P$ divide the line segment $AB$ internally in the ratio $7:3$. Let $3x - 25 = 0$ be a directrix of the ellipse $E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ and the corresponding focus be $S$. If from $S$, the perpendicular on the $x$-axis passes through $P$, then the length of the latus rectum of $E$ is equal to
(1) $\frac{25}{3}$
(2) $\frac{32}{9}$
(3) $\frac{25}{9}$
(4) $\frac{32}{5}$

Let $A B C D$ and $A E F G$ be squares of side 4 and 2 units, respectively. The point $E$ is on the line segment AB and the point F is on the diagonal AC . Then the radius r of the circle passing through the point F and touching the line segments BC and CD satisfies:
(1) $r = 0$
(2) $2 r ^ { 2 } - 4 r + 1 = 0$
(3) $2 r ^ { 2 } - 8 r + 7 = 0$
(4) $r ^ { 2 } - 8 r + 8 = 0$

If $\mathrm { P } ( 6,1 )$ be the orthocentre of the triangle whose vertices are $\mathrm { A } ( 5 , - 2 ) , \mathrm { B } ( 8,3 )$ and $\mathrm { C } ( \mathrm { h } , \mathrm { k } )$, then the point $C$ lies on the circle:
(1) $x ^ { 2 } + y ^ { 2 } - 61 = 0$
(2) $x ^ { 2 } + y ^ { 2 } - 52 = 0$
(3) $x ^ { 2 } + y ^ { 2 } - 65 = 0$
(4) $x ^ { 2 } + y ^ { 2 } - 74 = 0$

If the image of the point $( - 4,5 )$ in the line $x + 2 y = 2$ lies on the circle $( x + 4 ) ^ { 2 } + ( y - 3 ) ^ { 2 } = r ^ { 2 }$, then r is equal to: (1) 2 (2) 3 (3) 1 (4) 4

Let the shortest distance from $( \mathrm { a } , 0 )$, $\mathrm { a } > 0$, to the parabola $y ^ { 2 } = 4 x$ be 4. Then the equation of the circle passing through the point $( a , 0 )$ and the focus of the parabola, and having its centre on the axis of the parabola is :
(1) $x ^ { 2 } + y ^ { 2 } - 10 x + 9 = 0$
(2) $x ^ { 2 } + y ^ { 2 } - 6 x + 5 = 0$
(3) $x ^ { 2 } + y ^ { 2 } - 4 x + 3 = 0$
(4) $x ^ { 2 } + y ^ { 2 } - 8 x + 7 = 0$

The focus of the parabola $y ^ { 2 } = 4 x + 16$ is the centre of the circle $C$ of radius 5. If the values of $\lambda$, for which $C$ passes through the point of intersection of the lines $3 x - y = 0$ and $x + \lambda y = 4$, are $\lambda _ { 1 }$ and $\lambda _ { 2 }$, $\lambda _ { 1 } < \lambda _ { 2 }$, then $12 \lambda _ { 1 } + 29 \lambda _ { 2 }$ is equal to

Let circle $C$ be the image of $x^2 + y^2 - 2x + 4y - 4 = 0$ in the line $2x - 3y + 5 = 0$ and $A$ be the point on $C$ such that $OA$ is parallel to $x$-axis and $A$ lies on the right hand side of the centre $O$ of $C$. If $B(\alpha, \beta)$, with $\beta < 4$, lies on $C$ such that the length of the arc $AB$ is $(1/6)^{\text{th}}$ of the perimeter of $C$, then $\beta - \sqrt{3}\alpha$ is equal to
(1) $3 + \sqrt{3}$
(2) 4
(3) $4 - \sqrt{3}$
(4) 3

Let the circle $C$ touch the line $x - y + 1 = 0$, have the centre on the positive x-axis, and cut off a chord of length $\frac { 4 } { \sqrt { 13 } }$ along the line $- 3 x + 2 y = 1$. Let H be the hyperbola $\frac { x ^ { 2 } } { \alpha ^ { 2 } } - \frac { y ^ { 2 } } { \beta ^ { 2 } } = 1$, whose one of the foci is the centre of $C$ and the length of the transverse axis is the diameter of $C$. Then $2 \alpha ^ { 2 } + 3 \beta ^ { 2 }$ is equal to $\_\_\_\_$

Let the equation of the circle, which touches $x$-axis at the point $( a , 0 ) , a > 0$ and cuts off an intercept of length $b$ on $y$-axis be $x ^ { 2 } + y ^ { 2 } - \alpha x + \beta y + \gamma = 0$. If the circle lies below $x$-axis, then the ordered pair $( 2 a , b ^ { 2 } )$ is equal to
(1) $\left( \gamma , \beta ^ { 2 } - 4 \alpha \right)$
(2) $\left( \alpha , \beta ^ { 2 } + 4 \gamma \right)$
(3) $\left( \gamma , \beta ^ { 2 } + 4 \alpha \right)$
(4) $\left( \alpha , \beta ^ { 2 } - 4 \gamma \right)$

Let $y ^ { 2 } = 12 x$ be the parabola and $S$ be its focus. Let PQ be a focal chord of the parabola such that $( \mathrm { SP } ) ( \mathrm { SQ } ) = \frac { 147 } { 4 }$. Let C be the circle described taking PQ as a diameter. If the equation of a circle $C$ is $64 x ^ { 2 } + 64 y ^ { 2 } - \alpha x - 64 \sqrt { 3 } y = \beta$, then $\beta - \alpha$ is equal to $\_\_\_\_$ .

Consider three distinct points $A$, $B$, $C$ in the coordinate plane, where point $A$ is $(1, 1)$. Circles are drawn with line segments $\overline{AB}$ and $\overline{AC}$ as diameters. These two circles intersect at point $A$ and point $P(4, 2)$. Given that $\overline{PB} = 3\sqrt{10}$ and point $B$ is in the fourth quadrant, the coordinates of point $B$ are (（12），（13）（14）).

OAEF is a rectangle, ABCD is a square $| \mathrm { FE } | = 7$ units $| \mathrm { AB } | = 2$ units $| \mathrm { DE } | = x$
In the figure, points E and C are on a quarter circle with center O.
Accordingly, what is $x$ in units?
A) $\frac { 7 } { 2 }$ B) $\frac { 9 } { 2 }$ C) $\frac { 13 } { 4 }$ D) 3 E) 4

In the rectangular coordinate plane, a circle divided into two equal parts by the line $x + y = 4$ intersects the x-axis at a single point and the y-axis at two different points. Given that the distance between the points where the circle intersects the y-axis is 4 units, what is the circumference of the circle in units?
A) $4 \pi$
B) $5 \pi$
C) $6 \pi$
D) $7 \pi$
E) $8 \pi$

In a rectangular coordinate plane, point $A(11, 9)$ is located in the interior of a circle that is tangent to the line $y = x$ at point $B(7, 7)$.
Accordingly, what is the smallest integer value that the radius of this circle can take in units?
A) 6 B) 8 C) 10 D) 12 E) 14

Two identical blue ropes have one end each tied to two nails on a wall at equal heights from the ground and 48 units apart. Then a circular plate is hung on these ropes such that the other ends of the ropes are attached to two points on the circumference of the plate and the ropes are perpendicular to the ground, as shown in Figure 1. Later, one of these ropes broke and the plate hung on the remaining rope, and when the rope is perpendicular to the ground, the view in Figure 2 is obtained, and the height of the plate from the ground decreased by 16 units compared to the initial situation.
Accordingly, what is the radius of this plate in units?
A) 25 B) 26 C) 29 D) 30 E) 32