Let S be the focus of the hyperbola $\frac { x ^ { 2 } } { 3 } - \frac { y ^ { 2 } } { 5 } = 1$, on the positive $x$-axis. Let C be the circle with its centre at $A ( \sqrt { 6 } , \sqrt { 5 } )$ and passing through the point $S$. If $O$ is the origin and $S A B$ is a diameter of $C$, then the square of the area of the triangle OSB is equal to $\_\_\_\_$

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 $\_\_\_\_$

If the points of intersection of two distinct conics $x ^ { 2 } + y ^ { 2 } = 4 b$ and $\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ lie on the curve $y ^ { 2 } = 3 x ^ { 2 }$, then $3 \sqrt { 3 }$ times the area of the rectangle formed by the intersection points is $\_\_\_\_$ .

The area (in sq. units) of the part of circle $x ^ { 2 } + y ^ { 2 } = 169$ which is below the line $5 x - y = 13$ is $\frac { \pi \alpha } { 2 \beta } - \frac { 65 } { 2 } + \frac { \alpha } { \beta } \sin ^ { - 1 } \left( \frac { 12 } { 13 } \right)$ where $\alpha , \beta$ are coprime numbers. Then $\alpha + \beta$ 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 product of the focal distances of the point $\left(\sqrt{3}, \frac{1}{2}\right)$ on the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, $(a > b)$, be $\frac{7}{4}$. Then the absolute difference of the eccentricities of two such ellipses is
(1) $\frac{1 - \sqrt{3}}{\sqrt{2}}$
(2) $\frac{3 - 2\sqrt{2}}{2\sqrt{3}}$
(3) $\frac{3 - 2\sqrt{2}}{3\sqrt{2}}$
(4) $\frac{1 - 2\sqrt{2}}{\sqrt{3}}$

Let $ABCD$ be a trapezium whose vertices lie on the parabola $y ^ { 2 } = 4 x$. Let the sides $AD$ and $BC$ of the trapezium be parallel to y-axis. If the diagonal AC is of length $\frac { 25 } { 4 }$ and it passes through the point $( 1,0 )$, then the area of $ABCD$ is
(1) $\frac { 75 } { 4 }$
(2) $\frac { 25 } { 2 }$
(3) $\frac { 125 } { 8 }$
(4) $\frac { 75 } { 8 }$

The equation of the chord of the ellipse $\frac{x^{2}}{25} + \frac{y^{2}}{16} = 1$, whose mid-point is $(3, 1)$ is:
(1) $48x + 25y = 169$
(2) $5x + 16y = 31$
(3) $25x + 101y = 176$
(4) $4x + 122y = 134$

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

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 the parabola $y = x ^ { 2 } + \mathrm { p } x - 3$, meet the coordinate axes at the points $\mathrm { P } , \mathrm { Q }$ and R. If the circle C with centre at $( - 1 , - 1 )$ passes through the points $P , Q$ and $R$, then the area of $\triangle P Q R$ is:
(1) 7
(2) 4
(3) 6
(4) 5

If the line $3 x - 2 y + 12 = 0$ intersects the parabola $4 y = 3 x ^ { 2 }$ at the points $A$ and $B$, then at the vertex of the parabola, the line segment $A B$ subtends an angle equal to
(1) $\tan ^ { - 1 } \left( \frac { 4 } { 5 } \right)$
(2) $\tan ^ { - 1 } \left( \frac { 9 } { 7 } \right)$
(3) $\tan ^ { - 1 } \left( \frac { 11 } { 9 } \right)$
(4) $\frac { \pi } { 2 } - \tan ^ { - 1 } \left( \frac { 3 } { 2 } \right)$

Let $\mathrm { P } ( 4,4 \sqrt { 3 } )$ be a point on the parabola $y ^ { 2 } = 4 \mathrm { a } x$ and PQ be a focal chord of the parabola. If M and $N$ are the foot of perpendiculars drawn from $P$ and $Q$ respectively on the directrix of the parabola, then the area of the quadrilateral PQMN is equal to :
(1) $17 \sqrt { 3 }$
(2) $\frac { 263 \sqrt { 3 } } { 8 }$
(3) $\frac { 34 \sqrt { 3 } } { 3 }$
(4) $\frac { 343 \sqrt { 3 } } { 8 }$

The length of the chord of the ellipse $\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 2 } = 1$, whose mid-point is $\left( 1 , \frac { 1 } { 2 } \right)$, is :
(1) $\frac { 5 } { 3 } \sqrt { 15 }$
(2) $\frac { 1 } { 3 } \sqrt { 15 }$
(3) $\frac { 2 } { 3 } \sqrt { 15 }$
(4) $\sqrt { 15 }$

If $\alpha x + \beta y = 109$ is the equation of the chord of the ellipse $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 4 } = 1$, whose mid point is $\left( \frac { 5 } { 2 } , \frac { 1 } { 2 } \right)$, then $\alpha + \beta$ is equal to :
(1) 58
(2) 46
(3) 37
(4) 72

Let a circle $C$ pass through the points $( 4,2 )$ and $( 0,2 )$, and its centre lie on $3 x + 2 y + 2 = 0$. Then the length of the chord, of the circle $C$, whose mid-point is $( 1,2 )$, is :
(1) $\sqrt { 3 }$
(2) $2 \sqrt { 2 }$
(3) $2 \sqrt { 3 }$
(4) $4 \sqrt { 2 }$

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

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) $x ^ { 2 } + y ^ { 2 } - 8 x + 7 = 0$

If the midpoint of a chord of the ellipse $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 4 } = 1$ is $( \sqrt { 2 } , 4 / 3 )$, and the length of the chord is $\frac { 2 \sqrt { \alpha } } { 3 }$, then $\alpha$ is :
(1) 20
(2) 22
(3) 18
(4) 26

If the equation of the parabola with vertex $\mathrm{V}\left(\frac{3}{2}, 3\right)$ and the directrix $x + 2y = 0$ is $\alpha x^{2} + \beta y^{2} - \gamma xy - 30x - 60y + 225 = 0$, then $\alpha + \beta + \gamma$ is equal to:
(1) 7
(2) 9
(3) 8
(4) 6

Let the line $x + y = 1$ meet the circle $x^2 + y^2 = 4$ at the points A and B. If the line perpendicular to $AB$ and passing through the mid point of the chord $AB$ intersects the circle at $C$ and $D$, then the area of the quadrilateral ADBC is equal to:
(1) $\sqrt{14}$
(2) $3\sqrt{7}$
(3) $2\sqrt{14}$
(4) $5\sqrt{7}$

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 A and B be the two points of intersection of the line $y + 5 = 0$ and the mirror image of the parabola $y ^ { 2 } = 4 x$ with respect to the line $x + y + 4 = 0$. If d denotes the distance between A and B, and a denotes the area of $\triangle S A B$, where $S$ is the focus of the parabola $y ^ { 2 } = 4 x$, then the value of $( a + d )$ is

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 $y ^ { 2 } = 12 x$ 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 $\_\_\_\_$ .