Let $O(0,0)$ and $A(0,1)$ be two fixed points. Then, the locus of a point $P$ such that the perimeter of $\triangle AOP$ is 4 is
(1) $8x^2 + 9y^2 - 9y = 18$
(2) $9x^2 - 8y^2 + 8y = 16$
(3) $8x^2 - 9y^2 + 9y = 18$
(4) $9x^2 + 8y^2 - 8y = 16$

If one end of a focal chord of the parabola, $y ^ { 2 } = 16 x$ is at $( 1,4 )$, then the length of this focal chord is
(1) 24
(2) 25
(3) 22
(4) 20

The common tangent to the circles $x ^ { 2 } + y ^ { 2 } = 4$ and $x ^ { 2 } + y ^ { 2 } + 6 x + 8 y - 24 = 0$ also passes through the point:
(1) $( 4 , - 2 )$
(2) $( - 4,6 )$
(3) $( 6 , - 2 )$
(4) $( - 6,4 )$

If the circles $x^2 + y^2 - 16x - 20y + 164 = r^2$ and $(x-4)^2 + (y-7)^2 = 36$ intersect at two distinct points, then:
(1) $r > 11$
(2) $0 < r < 1$
(3) $1 < r < 11$
(4) $r = 11$

Let $S$ and $S ^ { \prime }$ be the foci of an ellipse and $B$ be any one of the extremities of its minor axis. If $\Delta S ^ { \prime } B S$ is a right angled triangle with right angle at $B$ and area $\left( \Delta S ^ { \prime } B S \right) = 8$ sq. units, then the length of a latus rectum of the ellipse is :
(1) $2 \sqrt { 2 }$
(2) 2
(3) 4
(4) $4 \sqrt { 2 }$

If the tangents on the ellipse $4x^2 + y^2 = 8$ at the points $(1,2)$ and $(a,b)$ are perpendicular to each other, then $a^2$ is equal to
(1) $\frac{2}{17}$
(2) $\frac{4}{17}$
(3) $\frac{64}{17}$
(4) $\frac{128}{17}$

The area (in sq. units) of the smaller of the two circles that touch the parabola, $y ^ { 2 } = 4 x$ at the point $( 1,2 )$ and the $x$-axis is
(1) $8 \pi ( 3 - 2 \sqrt { 2 } )$
(2) $8 \pi ( 2 - \sqrt { 2 } )$
(3) $4 \pi ( 3 + \sqrt { 2 } )$
(4) $4 \pi ( 2 - \sqrt { 2 } )$

If a line $y = mx + c$ is a tangent to the circle $(x - 3)^{2} + y^{2} = 1$, and it is perpendicular to a line $L_{1}$, where $L_{1}$ is the tangent to the circle $x^{2} + y^{2} = 1$ at the point $\left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)$, then
(1) $c^{2} - 7c + 6 = 0$
(2) $c^{2} + 7c + 6 = 0$
(3) $c^{2} + 6c + 7 = 0$
(4) $c^{2} - 6c + 7 = 0$

If the common tangent to the parabolas, $y ^ { 2 } = 4 x$ and $x ^ { 2 } = 4 y$ also touches the circle, $x ^ { 2 } + y ^ { 2 } = c ^ { 2 }$, then $c$ is equal to :
(1) $\frac { 1 } { 2 \sqrt { 2 } }$
(2) $\frac { 1 } { \sqrt { 2 } }$
(3) $\frac { 1 } { 4 }$
(4) $\frac { 1 } { 2 }$

If $y = m x + 4$ is a tangent to both the parabolas, $y ^ { 2 } = 4 x$ and $x ^ { 2 } = 2 b y$, then $b$ is equal to
(1) $-32$
(2) $-64$
(3) $-128$
(4) 128

A circle touches the $y$-axis at the point $( 0,4 )$ and passes through the point $( 2,0 )$. Which of the following lines is not a tangent to this circle?
(1) $4 x - 3 y + 17 = 0$
(2) $3 x - 4 y - 24 = 0$
(3) $3 x + 4 y - 6 = 0$
(4) $4 x + 3 y - 8 = 0$

Let P be a point on the parabola, $y ^ { 2 } = 12 x$ and N be the foot of the perpendicular drawn from $P$, on the axis of the parabola. A line is now drawn through the mid-point $M$ of $P N$, parallel to its axis which meets the parabola at $Q$. If the $y$-intercept of the line NQ is $\frac { 4 } { 3 }$, then:
(1) $P N = 4$
(2) $M Q = \frac { 1 } { 3 }$
(3) $M Q = \frac { 1 } { 4 }$
(4) $P N = 3$

Let the latus rectum of the parabola $y ^ { 2 } = 4 x$ be the common chord to the circles $C _ { 1 }$ and $C _ { 2 }$ each of them having radius $2 \sqrt { 5 }$. Then, the distance between the centres of the circles $C _ { 1 }$ and $C _ { 2 }$ is :
(1) 12
(2) 8
(3) $8 \sqrt { 5 }$
(4) $4 \sqrt { 5 }$

The circle passing through the intersection of the circles, $x ^ { 2 } + y ^ { 2 } - 6 x = 0$ and $x ^ { 2 } + y ^ { 2 } - 4 y = 0$ having its centre on the line, $2 x - 3 y + 12 = 0$, also passes through the point:
(1) $( - 1,3 )$
(2) $( - 3,6 )$
(3) $( - 3,1 )$
(4) $( 1 , - 3 )$

If the co-ordinates of two points $A$ and $B$ are $( \sqrt { 7 } , 0 )$ and $( - \sqrt { 7 } , 0 )$ respectively and $P$ is any point on the conic, $9 x ^ { 2 } + 16 y ^ { 2 } = 144$, then $PA + PB$ is equal to :
(1) 16
(2) 8
(3) 6
(4) 9

The centre of the circle passing through the point $(0,1)$ and touching the parabola $y=x^{2}$ at the point $(2,4)$ is
(1) $\left(\frac{-53}{10},\frac{16}{5}\right)$
(2) $\left(\frac{6}{5},\frac{53}{10}\right)$
(3) $\left(\frac{3}{10},\frac{16}{5}\right)$
(4) $\left(\frac{-16}{5},\frac{53}{10}\right)$

If one end of a focal chord $AB$ of the parabola $y ^ { 2 } = 8 x$ is at $A \left( \frac { 1 } { 2 } , - 2 \right)$, then the equation of the tangent to it at $B$ is:
(1) $2 x + y - 24 = 0$
(2) $x - 2 y + 8 = 0$
(3) $x + 2 y + 8 = 0$
(4) $2 x - y - 24 = 0$

If the point $P$ on the curve, $4 x ^ { 2 } + 5 y ^ { 2 } = 20$ is farthest from the point $Q ( 0 , - 4 )$, then $PQ ^ { 2 }$ is equal to
(1) 36
(2) 48
(3) 21
(4) 29

If the length of the chord of the circle, $x^2 + y^2 = r^2$ $(r > 0)$ along the line, $y - 2x = 3$ is $r$, then $r^2$ is equal to:
(1) $\frac{9}{5}$
(2) 12
(3) $\frac{24}{5}$
(4) $\frac{12}{5}$

Let $L _ { 1 }$ be a tangent to the parabola $y ^ { 2 } = 4 ( x + 1 )$ and $L _ { 2 }$ be a tangent to the parabola $y ^ { 2 } = 8 ( x + 2 )$ such that $L _ { 1 }$ and $L _ { 2 }$ intersect at right angles. Then $L _ { 1 }$ and $L _ { 2 }$ meet on the straight line:
(1) $x + 3 = 0$
(2) $2 x + 1 = 0$
(3) $x + 2 = 0$
(4) $x + 2 y = 0$

Let the tangents drawn from the origin to the circle, $x ^ { 2 } + y ^ { 2 } - 8 x - 4 y + 16 = 0$ touch it at the points $A$ and $B$. Then $( A B ) ^ { 2 }$ is equal to
(1) $\frac { 52 } { 5 }$
(2) $\frac { 56 } { 5 }$
(3) $\frac { 64 } { 5 }$
(4) $\frac { 32 } { 5 }$

The area (in sq. units) of an equilateral triangle inscribed in the parabola $y ^ { 2 } = 8 x$, with one of its vertices on the vertex of this parabola is
(1) $64 \sqrt { 3 }$
(2) $256 \sqrt { 3 }$
(3) $192 \sqrt { 3 }$
(4) $128 \sqrt { 3 }$

Which of the following points lies on the locus of the foot of perpendicular drawn upon any tangent to the ellipse, $\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 2 } = 1$ from any of its foci?
(1) $( - 2 , \sqrt { 3 } )$
(2) $( - 1 , \sqrt { 2 } )$
(3) $( - 1 , \sqrt { 3 } )$
(4) $( 1,2 )$

Let $PQ$ be a diameter of the circle $x ^ { 2 } + y ^ { 2 } = 9$. If $\alpha$ and $\beta$ are the lengths of the perpendiculars from $P$ and $Q$ on the straight line, $x + y = 2$ respectively, then the maximum value of $\alpha \beta$ is $\_\_\_\_$

Let the lines $( 2 - i ) z = ( 2 + i ) \bar { z }$ and $( 2 + i ) z + ( i - 2 ) \bar { z } - 4 i = 0$, (here $i ^ { 2 } = - 1$ ) be normal to a circle $C$. If the line $i z + \bar { z } + 1 + i = 0$ is tangent to this circle $C$, then its radius is :
(1) $\frac { 3 } { \sqrt { 2 } }$
(2) $3 \sqrt { 2 }$
(3) $\frac { 3 } { 2 \sqrt { 2 } }$
(4) $\frac { 1 } { 2 \sqrt { 2 } }$