Let the tangents at the points $A ( 4 , - 11 )$ and $B ( 8 , - 5 )$ on the circle $x ^ { 2 } + y ^ { 2 } - 3 x + 10 y - 15 = 0$, intersect at the point $C$. Then the radius of the circle, whose centre is $C$ and the line joining $A$ and $B$ is its tangent, is equal to
(1) $\frac { 3 \sqrt { 3 } } { 4 }$
(2) $2 \sqrt { 13 }$
(3) $\sqrt { 13 }$
(4) $\frac { 2 \sqrt { 13 } } { 3 }$

A triangle is formed by the tangents at the point $( 2,2 )$ on the curves $y ^ { 2 } = 2 x$ and $x ^ { 2 } + y ^ { 2 } = 4 x$, and the line $\mathrm { x } + \mathrm { y } + 2 = 0$. If $r$ is the radius of its circumcircle, then $r ^ { 2 }$ is equal to $\_\_\_\_$

In a group of 100 persons 75 speak English and 40 speak Hindi. Each person speaks at least one of the two languages. If the number of persons who speak only English is $\alpha$ and the number of persons who speaks only Hindi is $\beta$, then the eccentricity of the ellipse $25\left(\beta^{2}x^{2} + \alpha^{2}y^{2}\right) = \alpha^{2}\beta^{2}$ is
(1) $\frac{\sqrt{119}}{12}$
(2) $\frac{\sqrt{117}}{12}$
(3) $\frac{3\sqrt{15}}{12}$
(4) $\frac{\sqrt{129}}{12}$

The number of integral values of $k$ for which the line $3x + 4y = k$ intersects the circle $x^2 + y^2 - 2x - 4y + 4 = 0$ at two distinct points is $\_\_\_\_$.

Let a common tangent to the curves $y^2 = 4x$ and $(x-4)^2 + y^2 = 16$ touch the curves at the points $P$ and $Q$. Then $PQ^2$ is equal to $\_\_\_\_$.

If the circles $( x + 1 ) ^ { 2 } + ( y + 2 ) ^ { 2 } = r ^ { 2 }$ and $x ^ { 2 } + y ^ { 2 } - 4 x - 4 y + 4 = 0$ intersect at exactly two distinct points, then
(1) $5 < \mathrm { r } < 9$
(2) $0 < \mathrm { r } < 7$
(3) $3 < r < 7$
(4) $\frac { 1 } { 2 } < \mathrm { r } < 7$

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 $\left( 5 , \frac { a } { 4 } \right)$, be the circumcenter of a triangle with vertices $A ( a , - 2 ) , B ( a , 6 )$ and $C \left( \frac { a } { 4 } , - 2 \right)$. Let $\alpha$ denote the circumradius, $\beta$ denote the area and $\gamma$ denote the perimeter of the triangle. Then $\alpha + \beta + \gamma$ is
(1) 60
(2) 53
(3) 62
(4) 30

Let $C$ be a circle with radius $\sqrt { 10 }$ units and centre at the origin. Let the line $x + y = 2$ intersects the circle C at the points P and Q . Let MN be a chord of C of length 2 unit and slope - 1 . Then, a distance (in units) between the chord PQ and the chord MN is
(1) $3 - \sqrt { 2 }$
(2) $\sqrt { 2 } + 1$
(3) $\sqrt { 2 } - 1$
(4) $2 - \sqrt { 3 }$

Let the locus of the mid points of the chords of circle $x^2 + (y-1)^2 = 1$ drawn from the origin intersect the line $x + y = 1$ at $P$ and $Q$. Then, the length of $PQ$ is:
(1) $\frac{1}{\sqrt{2}}$
(2) $\sqrt{2}$
(3) $\frac{1}{2}$
(4) 1

A circle is inscribed in an equilateral triangle of side of length 12 . If the area and perimeter of any square inscribed in this circle are $m$ and $n$, respectively, then $m + n ^ { 2 }$ is equal to
(1) 408
(2) 414
(3) 396
(4) 312

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) $r ^ { 2 } - 8 r + 8 = 0$

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

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 a variable line passing through the centre of the circle $x^2 + y^2 - 16x - 4y = 0$, meet the positive coordinate axes at the point $A$ and $B$. Then the minimum value of $OA + OB$, where $O$ is the origin, is equal to
(1) 12
(2) 18
(3) 20
(4) 24

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) $( 2,4 )$

If the shortest distance of the parabola $y ^ { 2 } = 4 x$ from the centre of the circle $x ^ { 2 } + y ^ { 2 } - 4 x - 16 y + 64 = 0$ is $d$ , then $\mathrm { d } ^ { 2 }$ is equal to:
(1) 16
(2) 24
(3) 20
(4) 36

Let the circle $C _ { 1 } : x ^ { 2 } + y ^ { 2 } - 2 ( x + y ) + 1 = 0$ and $C _ { 2 }$ be a circle having centre at $( - 1,0 )$ and radius 2 . If the line of the common chord of $\mathrm { C } _ { 1 }$ and $\mathrm { C } _ { 2 }$ intersects the $y$-axis at the point P , then the square of the distance of P from the centre of $\mathrm { C } _ { 1 }$ is :
(1) 2
(2) 1
(3) 4
(4) 6

If the locus of the point, whose distances from the point $( 2,1 )$ and $( 1,3 )$ are in the ratio $5 : 4$, is $a x ^ { 2 } + b y ^ { 2 } + c x y + d x + e y + 170 = 0$, then the value of $a ^ { 2 } + 2 b + 3 c + 4 d + e$ is equal to:
(1) 37
(2) 437
(3) $- 27$
(4) 5

Let $A , B$ and $C$ be three points on the parabola $y ^ { 2 } = 6 x$ and let the line segment $A B$ meet the line $L$ through $C$ parallel to the $x$-axis at the point $D$. Let $M$ and $N$ respectively be the feet of the perpendiculars from $A$ and $B$ on $L$. Then $\left( \frac { A M \cdot B N } { C D } \right) ^ { 2 }$ is equal to $\_\_\_\_$

Equations of two diameters of a circle are $2 x - 3 y = 5$ and $3 x - 4 y = 7$. The line joining the points $\left( - \frac { 22 } { 7 } , - 4 \right)$ and $\left( - \frac { 1 } { 7 } , 3 \right)$ intersects the circle at only one point $P ( \alpha , \beta )$. Then $17 \beta - \alpha$ 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 $\_\_\_\_$ .