If the tangent at a point on the ellipse $\frac{x^2}{27} + \frac{y^2}{3} = 1$ meets the coordinate axes at $A$ and $B$, and $O$ is the origin, then the minimum area (in sq. units) of the triangle $OAB$ is:
(1) $\frac{9}{2}$
(2) $9$
(3) $9\sqrt{3}$
(4) $\frac{\sqrt{3}}{2}$

The radius of a circle, having minimum area, which touches the curve $y = 4 - x ^ { 2 }$ and the lines $y = | x |$ is:
(1) $2 ( \sqrt { 2 } + 1 )$
(2) $2 ( \sqrt { 2 } - 1 )$
(3) $4 ( \sqrt { 2 } - 1 )$
(4) $4 ( \sqrt { 2 } + 1 )$

The locus of the point of intersection of the straight lines, $t x - 2 y - 3 t = 0$ and $x - 2 t y + 3 = 0 ( t \in R )$, is:
(1) A hyperbola with the length of conjugate axis 3
(2) A hyperbola with eccentricity $\sqrt { 5 }$
(3) An ellipse with the length of major axis 6
(4) An ellipse with eccentricity $\frac { 2 } { \sqrt { 5 } }$

The radius of a circle, having minimum area, which touches the curve $y = 4 - x^2$ and the lines $y = |x|$ is:
(1) $2(\sqrt{2} + 1)$
(2) $2(\sqrt{2} - 1)$
(3) $4(\sqrt{2} - 1)$
(4) $4(\sqrt{2} + 1)$

If two parallel chords of a circle, having diameter 4 units, lie on the opposite sides of the center and subtend angles $\cos ^ { - 1 } \left( \frac { 1 } { 7 } \right)$ and $\sec ^ { - 1 } ( 7 )$ at the center respectively, then the distance between these chords is:
(1) $\frac { 8 } { \sqrt { 7 } }$
(2) $\frac { 16 } { 7 }$
(3) $\frac { 4 } { \sqrt { 7 } }$
(4) $\frac { 8 } { 7 }$

If the common tangents to the parabola, $x ^ { 2 } = 4 y$ and the circle, $x ^ { 2 } + y ^ { 2 } = 4$ intersect at the point $P$, then the distance of $P$ from the origin (units), is:
(1) $2 ( 3 + 2 \sqrt { 2 } )$
(2) $3 + 2 \sqrt { 2 }$
(3) $\sqrt { 2 } + 1$
(4) $2 ( \sqrt { 2 } + 1 )$

If a point $P ( 0 , - 2 )$ and $Q$ is any point on the circle, $x ^ { 2 } + y ^ { 2 } - 5 x - y + 5 = 0$, then the maximum value of $( P Q ) ^ { 2 }$ is
(1) $8 + 5 \sqrt { 3 }$
(2) $\frac { 47 + 10 \sqrt { 6 } } { 2 }$
(3) $14 + 5 \sqrt { 3 }$
(4) $\frac { 25 + \sqrt { 6 } } { 2 }$

Consider an ellipse, whose center is at the origin and its major axis is along the $x$-axis. If its eccentricity is $\frac { 3 } { 5 }$ and the distance between its foci is 6, then the area (in sq. units) of the quadrilateral inscribed in the ellipse, with the vertices as the vertices of the ellipse, is:
(1) 32
(2) 80
(3) 40
(4) 8

A circle passes through the points $( 2,3 )$ and $( 4,5 )$. If its centre lies on the line $y - 4 x + 3 = 0$, then its radius is equal to :
(1) $\sqrt { 5 }$
(2) $\sqrt { 2 }$
(3) 2
(4) 1

If a circle $C$, whose radius is 3 , touches externally the circle $x ^ { 2 } + y ^ { 2 } + 2 x - 4 y - 4 = 0$ at the point $( 2,2 )$, then the length of the intercept cut by this circle $C$ on the $x$-axis is equal to
(1) $2 \sqrt { 3 }$
(2) $\sqrt { 5 }$
(3) $3 \sqrt { 2 }$
(4) $2 \sqrt { 5 }$

If the tangent at $( 1,7 )$ to the curve $x ^ { 2 } = y - 6$ touch the circle $x ^ { 2 } + y ^ { 2 } + 16 x + 12 y + c = 0$ then the value of $c$ is:
(1) 95
(2) 195
(3) 185
(4) 85

Two parabolas with a common vertex and with axes along the $x$-axis and $y$-axis respectively, intersect each other in the first quadrant. If the length of the latus rectum of each parabola is 3, then the equation of the common tangent to the two parabolas is :
(1) $3 ( x + y ) + 4 = 0$
(2) $8 ( 2 x + y ) + 3 = 0$
(3) $x + 2 y + 3 = 0$
(4) $4 ( x + y ) + 3 = 0$

A circle passes through the points $( 2,3 )$ and $( 4,5 )$. If its centre lies on the line, $y - 4 x + 3 = 0$, then its radius is equal to
(1) $\sqrt { 5 }$
(2) 1
(3) $\sqrt { 2 }$
(4) 2

Let $P$ be a point on the parabola $x ^ { 2 } = 4 y$. If the distance of $P$ from the center of the circle $x ^ { 2 } + y ^ { 2 } + 6 x + 8 = 0$ is minimum, then the equation of the tangent to the parabola at $P$ is
(1) $x + y + 1 = 0$
(2) $x + 4 y - 2 = 0$
(3) $x + 2 y = 0$
(4) $x - y + 3 = 0$

Tangent and normal are drawn at $P ( 16,16 )$ on the parabola $y ^ { 2 } = 16 x$, which intersect the axis of the parabola at $A \& B$, respectively. If $C$ is the center of the circle through the points $P , A \& B$ and $\angle C P B = \theta$, then a value of $\tan \theta$ is:
(1) $\frac { 4 } { 3 }$
(2) $\frac { 1 } { 2 }$
(3) 2
(4) 3

Two parabolas with a common vertex and with axes along $x$-axis and $y$-axis, respectively, intersect each other in the first quadrant. if the length of the latus rectum of each parabola is 3 , then the equation of the common tangent to the two parabolas is?
(1) $3 ( x + y ) + 4 = 0$
(2) $8 ( 2 x + y ) + 3 = 0$
(3) $4 ( x + y ) + 3 = 0$
(4) $x + 2 y + 3 = 0$

Two sets $A$ and $B$ are as under: $A = \{ ( a , b ) \in R \times R : | a - 5 | < 1$ and $| b - 5 | < 1 \}$; $B = \left\{ ( a , b ) \in R \times R : 4 ( a - 6 ) ^ { 2 } + 9 ( b - 5 ) ^ { 2 } \leq 36 \right\}$. Then :
(1) neither $A \subset B$ nor $B \subset A$
(2) $B \subset A$
(3) $A \subset B$
(4) $A \cap B = \phi$ (an empty set)

All the points in the set $S = \left\{ \frac { \alpha + i } { \alpha - i } , \alpha \in R \right\} , i = \sqrt { - 1 }$ lie on a
(1) straight line whose slope is - 1
(2) circle whose radius is $\sqrt { 2 }$
(3) circle whose radius is 1
(4) straight line whose slope is 1

The tangent and the normal lines at the point $( \sqrt { 3 } , 1 )$ to the circle $x ^ { 2 } + y ^ { 2 } = 4$ and the $x$-axis form a triangle. The area of this triangle (in square units) is:
(1) $\frac { 1 } { 3 }$
(2) $\frac { 2 } { \sqrt { 3 } }$
(3) $\frac { 4 } { \sqrt { 3 } }$
(4) $\frac { 1 } { \sqrt { 3 } }$

If a circle of radius $R$ passes through the origin $O$ and intersects the coordinate axes at $A$ and $B$, then the locus of the foot of perpendicular from $O$ on $AB$ is :
(1) $\left( x ^ { 2 } + y ^ { 2 } \right) ( x + y ) = R ^ { 2 } x y$
(2) $\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 3 } = 4 R ^ { 2 } x ^ { 2 } y ^ { 2 }$
(3) $\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = 4 R ^ { 2 } x ^ { 2 } y ^ { 2 }$
(4) $\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = 4 R x ^ { 2 } y ^ { 2 }$

The sum of the squares of the lengths of the chords intercepted on the circle, $x^2 + y^2 = 16$, by the lines, $x + y = n$, $n \in N$, where $N$ is the set of all natural numbers is:
(1) 210
(2) 105
(3) 320
(4) 160

If a tangent to the circle $x ^ { 2 } + y ^ { 2 } = 1$ intersects the coordinate axes at distinct points $P$ and $Q$, then the locus of the mid-point of $PQ$ is:
(1) $x ^ { 2 } + y ^ { 2 } - 16 x ^ { 2 } y ^ { 2 } = 0$
(2) $x ^ { 2 } + y ^ { 2 } - 4 x ^ { 2 } y ^ { 2 } = 0$
(3) $x ^ { 2 } + y ^ { 2 } - 2 x y = 0$
(4) $x ^ { 2 } + y ^ { 2 } - 2 x ^ { 2 } y ^ { 2 } = 0$

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) $x = y \cot \theta - 2 \tan \theta$