A rectangle is inscribed in a circle with a diameter lying along the line $3 y = x + 7$. If the two adjacent vertices of the rectangle are $( - 8,5 )$ and $( 6,5 )$, then the area of the rectangle (in sq. units) is:
(1) 72
(2) 98
(3) 56
(4) 84

If a circle $C$ passing through the point $( 4,0 )$ touches the circle $x ^ { 2 } + y ^ { 2 } + 4 x - 6 y = 12$ externally at the point $( 1 , - 1 )$, then the radius of $C$ is:
(1) 4 units
(2) 5 units
(3) $2 \sqrt { 5 }$ units
(4) $\sqrt { 57 }$ units

The equation of a tangent to the parabola, $x ^ { 2 } = 8 y$, which makes an angle $\theta$ with the positive direction of $x$-axis, is
(1) $y = x \tan \theta + 2 \cot \theta$
(2) $y = x \tan \theta - 2 \cot \theta$
(3) $x = y \cot \theta + 2 \tan \theta$
(4) $x = y \cot \theta - 2 \tan \theta$

In an ellipse, with centre at the origin, if the difference of the lengths of major axis and minor axis is 10 and one of the foci is at $( 0 , 5 \sqrt { 3 } )$, then the length of its latus rectum is:
(1) 6
(2) 10
(3) 8
(4) 5

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) $9x^2 + 8y^2 - 8y = 16$

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) $r = 11$

If the parabolas $y ^ { 2 } = 4 b ( x - c )$ and $y ^ { 2 } = 8 a x$ have a common normal, then which one of the following is a valid choice for the ordered triad $( a , b , c )$
(1) $( 1,1,3 )$
(2) $\left( \frac { 1 } { 2 } , 2,0 \right)$
(3) $\left( \frac { 1 } { 2 } , 2,3 \right)$
(4) All of above

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 eccentricity of the standard hyperbola passing through the point $( 4 , 6 )$ is 2 , then the equation of the tangent to the hyperbola at $( 4 , 6 )$ is:
(1) $2 x - 3 y + 10 = 0$
(2) $x - 2 y + 8 = 0$
(3) $3 x - 2 y = 0$
(4) $2 x - y - 2 = 0$

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 the tangent to the parabola $y ^ { 2 } = x$ at a point $( \alpha , \beta ) , ( \beta > 0 )$ is also a tangent to the ellipse, $x ^ { 2 } + 2 y ^ { 2 } = 1$ then $\alpha$ is equal to:
(1) $\sqrt { 2 } - 1$
(2) $2 \sqrt { 2 } + 1$
(3) $\sqrt { 2 } + 1$
(4) $2 \sqrt { 2 } - 1$

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) $\frac { 1 } { 2 }$

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 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) $( 1 , - 3 )$

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 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 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) $\frac { 32 } { 5 }$

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) $\frac { 1 } { 2 \sqrt { 2 } }$

If the locus of the mid-point of the line segment from the point $( 3,2 )$ to a point on the circle, $x ^ { 2 } + y ^ { 2 } = 1$ is a circle of radius $r$, then $r$ is equal to
(1) $\frac { 1 } { 4 }$
(2) 1
(3) $\frac { 1 } { 3 }$
(4) $\frac { 1 } { 2 }$

Let the lengths of intercepts on $x$-axis and $y$-axis made by the circle $x ^ { 2 } + y ^ { 2 } + ax + 2ay + c = 0 , ( a < 0 )$ be $2 \sqrt { 2 }$ and $2 \sqrt { 5 }$, respectively. Then the shortest distance from origin to a tangent to this circle which is perpendicular to the line $x + 2y = 0$, is equal to :
(1) $\sqrt { 11 }$
(2) $\sqrt { 7 }$
(3) $\sqrt { 6 }$
(4) $\sqrt { 10 }$