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

Let a circle $C$ touch the lines $L _ { 1 } : 4 x - 3 y + K _ { 1 } = 0$ and $L _ { 2 } : 4 x - 3 y + K _ { 2 } = 0 , K _ { 1 } , \quad K _ { 2 } \in R$. If a line passing through the centre of the circle $C$ intersects $L _ { 1 }$ at $( -1, 2 )$ and $L _ { 2 }$ at $( 3 , - 6 )$, then the equation of the circle $C$ is
(1) $( x - 1 ) ^ { 2 } + ( y - 2 ) ^ { 2 } = 4$
(2) $( x - 1 ) ^ { 2 } + ( y + 2 ) ^ { 2 } = 16$
(3) $( x + 1 ) ^ { 2 } + ( y - 2 ) ^ { 2 } = 4$
(4) $( x - 1 ) ^ { 2 } + ( y - 2 ) ^ { 2 } = 16$

Let the tangents at two points $A$ and $B$ on the circle $x ^ { 2 } + y ^ { 2 } - 4 x + 3 = 0$ meet at origin $O ( 0,0 )$. Then the area of the triangle of $O A B$ is
(1) $\frac { 3 \sqrt { 3 } } { 2 }$
(2) $\frac { 3 \sqrt { 3 } } { 4 }$
(3) $\frac { 3 } { 2 \sqrt { 3 } }$
(4) $\frac { 3 } { 4 \sqrt { 3 } }$

If $y = m _ { 1 } x + c _ { 1 }$ and $y = m _ { 2 } x + c _ { 2 } , \quad m _ { 1 } \neq m _ { 2 }$ are two common tangents of circle $x ^ { 2 } + y ^ { 2 } = 2$ and parabola $y ^ { 2 } = x$, then the value of $8 \quad m _ { 1 } \quad m _ { 2 }$ is equal to
(1) $3 \sqrt { 2 } - 4$
(2) $6 \sqrt { 2 } - 4$
(3) $- 5 + 6 \sqrt { 2 }$
(4) $3 + 4 \sqrt { 2 }$

If the tangents drawn at the point $O(0,0)$ and $P(1+\sqrt{5}, 2)$ on the circle $x^2 + y^2 - 2x - 4y = 0$ intersect at the point $Q$, then the area of the triangle $OPQ$ is equal to
(1) $\frac{3+\sqrt{5}}{2}$
(2) $\frac{4+2\sqrt{5}}{2}$
(3) $\frac{5+3\sqrt{5}}{2}$
(4) $\frac{7+3\sqrt{5}}{2}$

Let the abscissae of the two points $P$ and $Q$ on a circle be the roots of $x ^ { 2 } - 4 x - 6 = 0$ and the ordinates of $P$ and $Q$ be the roots of $y ^ { 2 } + 2 y - 7 = 0$. If $PQ$ is a diameter of the circle $x ^ { 2 } + y ^ { 2 } + 2 a x + 2 b y + c = 0$, then the value of $a + b - c$ is
(1) 12
(2) 13
(3) 14
(4) 16

Let $C$ be a circle passing through the points $A ( 2 , - 1 )$ and $B ( 3,4 )$. The line segment $AB$ is not a diameter of $C$. If $r$ is the radius of $C$ and its centre lies on the circle $( x - 5 ) ^ { 2 } + ( y - 1 ) ^ { 2 } = \frac { 13 } { 2 }$, then $r ^ { 2 }$ is equal to
(1) 32
(2) $\frac { 65 } { 2 }$
(3) $\frac { 61 } { 2 }$
(4) 30

A point $P$ moves so that the sum of squares of its distances from the points $( 1,2 )$ and $( - 2,1 )$ is 14. Let $f ( x , y ) = 0$ be the locus of $P$, which intersects the $x$-axis at the points $A , B$ and the $y$-axis at the point $C , D$. Then the area of the quadrilateral $ACBD$ is equal to
(1) $\frac { 9 } { 2 }$
(2) $\frac { 3 \sqrt { 17 } } { 2 }$
(3) $\frac { 3 \sqrt { 17 } } { 4 }$
(4) 9

A particle is moving in the $x y$-plane along a curve $C$ passing through the point $( 3,3 )$. The tangent to the curve $C$ at the point $P$ meets the $x$-axis at $Q$. If the $y$-axis bisects the segment $P Q$, then $C$ is a parabola with
(1) length of latus rectum 3
(2) length of latus rectum 6
(3) focus $\left( \frac { 4 } { 3 } , 0 \right)$
(4) focus $\left( 0 , \frac { 3 } { 3 } \right)$

Let the tangent to the circle $C _ { 1 } : x ^ { 2 } + y ^ { 2 } = 2$ at the point $M ( - 1,1 )$ intersect the circle $C _ { 2 }$ : $( x - 3 ) ^ { 2 } + ( y - 2 ) ^ { 2 } = 5$, at two distinct points $A$ and $B$. If the tangents to $C _ { 2 }$ at the points $A$ and $B$ intersect at $N$, then the area of the triangle $A N B$ is equal to
(1) $\frac { 1 } { 2 }$
(2) $\frac { 2 } { 3 }$
(3) $\frac { 1 } { 6 }$
(4) $\frac { 5 } { 3 }$

The equation of a common tangent to the parabolas $y = x ^ { 2 }$ and $y = -(x - 2) ^ { 2 }$ is
(1) $y = 4 x - 2$
(2) $y = 4 x - 1$
(3) $y = 4 x + 1$
(4) $y = 4 x + 2$

Let the normal at the point $P$ on the parabola $y ^ { 2 } = 6 x$ pass through the point $( 5 , - 8 )$. If the tangent at $P$ to the parabola intersects its directrix at the point $Q$, then the ordinate of the point $Q$ is
(1) $\frac { - 9 } { 4 }$
(2) $\frac { 9 } { 4 }$
(3) $\frac { - 5 } { 2 }$
(4) $- 3$

Let the locus of the centre $(\alpha , \beta),\ \beta > 0$, of the circle which touches the circle $x ^ { 2 } + (y - 1) ^ { 2 } = 1$ externally and also touches the $x$-axis be $L$. Then the area bounded by $L$ and the line $y = 4$ is
(1) $\frac { 32 \sqrt { 2 } } { 3 }$
(2) $\frac { 40 \sqrt { 2 } } { 3 }$
(3) $\frac { 64 } { 3 }$
(4) $\frac { 32 } { 3 }$

If the circle $x ^ { 2 } + y ^ { 2 } - 2 g x + 6 y - 19 c = 0 , g , c \in \mathbb { R }$ passes through the point $( 6,1 )$ and its centre lies on the line $x - 2 c y = 8$, then the length of intercept made by the circle on $x$-axis is
(1) $\sqrt { 11 }$
(2) 4
(3) 3
(4) $2 \sqrt { 23 }$

Let $x ^ { 2 } + y ^ { 2 } + A x + B y + C = 0$ be a circle passing through ( 0,6 ) and touching the parabola $y = x ^ { 2 }$ at ( 2,4 ). Then $A + C$ is equal to
(1) 16
(2) $\frac { 88 } { 5 }$
(3) 72
(4) - 8

The tangents at the points $A(1,3)$ and $B(1,-1)$ on the parabola $y^2 - 2x - 2y = 1$ meet at the point $P$. Then the area (in unit$^2$) of the triangle $PAB$ is:
(1) 4
(2) 6
(3) 7
(4) 8

Let $C$ be the centre of the circle $x^2 + y^2 - x + 2y = \frac{11}{4}$ and $P$ be a point on the circle. A line passes through the point $C$, makes an angle of $\frac{\pi}{4}$ with the line $CP$ and intersects the circle at the points $Q$ and $R$. Then the area of the triangle $PQR$ (in unit$^2$) is
(1) 2
(2) $2\sqrt{2}$
(3) $8\sin\frac{\pi}{8}$
(4) $8\cos\frac{\pi}{8}$

The set of values of $k$ for which the circle $C : 4 x ^ { 2 } + 4 y ^ { 2 } - 12 x + 8 y + k = 0$ lies inside the fourth quadrant and the point $\left( 1 , - \frac { 1 } { 3 } \right)$ lies on or inside the circle $C$ is
(1) An empty set
(2) $\left( 6 , \frac { 95 } { 9 } \right]$
(3) $\left[ \frac { 80 } { 9 } , 10 \right)$
(4) $\left( 9 , \frac { 92 } { 9 } \right]$

Let $P ( a , b )$ be a point on the parabola $y ^ { 2 } = 8 x$ such that the tangent at $P$ passes through the centre of the circle $x ^ { 2 } + y ^ { 2 } - 10 x - 14 y + 65 = 0$. Let $A$ be the product of all possible values of $a$ and $B$ be the product of all possible values of $b$. Then the value of $A + B$ is equal to
(1) 0
(2) 25
(3) 40
(4) 65

A circle $C _ { 1 }$ passes through the origin $O$ and has diameter 4 on the positive $x$-axis. The line $y = 2 x$ gives a chord $O A$ of a circle $C _ { 1 }$. Let $C _ { 2 }$ be the circle with $O A$ as a diameter. If the tangent to $C _ { 2 }$ at the point $A$ meets the $x$-axis at $P$ and $y$-axis at $Q$, then $Q A : A P$ is equal to
(1) $1 : 4$
(2) $1 : 5$
(3) $2 : 5$
(4) $1 : 3$

A circle touches both the $y$-axis and the line $x + y = 0$. Then the locus of its center is
(1) $y = \sqrt{2}x$
(2) $x = \sqrt{2}y$
(3) $y^2 - x^2 = 2xy$
(4) $x^2 - y^2 = 2xy$

Let $A(\alpha, -2)$, $B(\alpha, 6)$ and $C\left(\frac{\alpha}{4}, -2\right)$ be vertices of a $\triangle ABC$. If $\left(5, \frac{\alpha}{4}\right)$ is the circumcentre of $\triangle ABC$, then which of the following is NOT correct about $\triangle ABC$
(1) area is 24
(2) perimeter is 25
(3) circumradius is 5
(4) inradius is 2

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

The line $y = x + 1$ meets the ellipse $\frac{x^2}{4} + \frac{y^2}{2} = 1$ at two points $P$ and $Q$. If $r$ is the radius of the circle with $PQ$ as diameter then $3r^2$ is equal to
(1) 20
(2) 12
(3) 11
(4) 8

Let $P$ and $Q$ be any points on the curves $( x - 1 ) ^ { 2 } + ( y + 1 ) ^ { 2 } = 1$ and $y = x ^ { 2 }$, respectively. The distance between $P$ and $Q$ is minimum for some value of the abscissa of $P$ in the interval
(1) $\left( 0 , \frac { 1 } { 4 } \right)$
(2) $\left( \frac { 1 } { 2 } , \frac { 3 } { 4 } \right)$
(3) $\left( \frac { 1 } { 4 } , \frac { 1 } { 2 } \right)$
(4) $\left( \frac { 3 } { 4 } , 1 \right)$