Let $P _ { 1 }$ be a parabola with vertex $( 3,2 )$ and focus $( 4,4 )$ and $P _ { 2 }$ be its mirror image with respect to the line $x + 2 y = 6$. Then the directrix of $P _ { 2 }$ is $x + 2 y =$ $\_\_\_\_$.

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

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 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

Let the equation of two diameters of a circle $x ^ { 2 } + y ^ { 2 } - 2x + 2fy + 1 = 0$ be $2px - y = 1$ and $2x + py = 4p$. Then the slope $m \in (0,\infty)$ of the tangent to the hyperbola $3x ^ { 2 } - y ^ { 2 } = 3$ passing through the centre of the circle is equal to $\_\_\_\_$.

If the length of the latus rectum of the ellipse $x ^ { 2 } + 4 y ^ { 2 } + 2 x + 8 y - \lambda = 0$ is 4 , and $l$ is the length of its major axis, then $\lambda + l$ is equal to $\_\_\_\_$ .

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 $\_\_\_\_$ .

Q66. Let a circle $C$ of radius 1 and closer to the origin be such that the lines passing through the point $( 3,2 )$ and parallel to the coordinate axes touch it. Then the shortest distance of the circle C from the point $( 5,5 )$ is :
(1) $2 \sqrt { 2 }$
(2) $4 \sqrt { 2 }$
(3) 4
(4) 5