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 $\frac { 2 } { \sqrt { 5 } }$

If the tangent drawn to the hyperbola $4 y ^ { 2 } = x ^ { 2 } + 1$ intersect the co-ordinates axes at the distinct points $A$ and $B$, then the locus of the midpoint of $AB$ is :
(1) $x ^ { 2 } - 4 y ^ { 2 } + 16 x ^ { 2 } y ^ { 2 } = 0$
(2) $4 x ^ { 2 } - y ^ { 2 } + 16 x ^ { 2 } y ^ { 2 } = 0$
(3) $x ^ { 2 } - 4 y ^ { 2 } - 16 x ^ { 2 } y ^ { 2 } = 0$
(4) $4 x ^ { 2 } - y ^ { 2 } - 16 x ^ { 2 } y ^ { 2 } = 0$

If the tangents drawn to the hyperbola $4 y ^ { 2 } = x ^ { 2 } + 1$ intersect the co-ordinate axes at the distinct points $A$ and $B$, then the locus of the mid point of $A B$ is
(1) $x ^ { 2 } - 4 y ^ { 2 } + 16 x ^ { 2 } y ^ { 2 } = 0$
(2) $4 x ^ { 2 } - y ^ { 2 } + 16 x ^ { 2 } y ^ { 2 } = 0$
(3) $4 x ^ { 2 } - y ^ { 2 } - 16 x ^ { 2 } y ^ { 2 } = 0$
(4) $x ^ { 2 } - 4 y ^ { 2 } - 16 x ^ { 2 } y ^ { 2 } = 0$

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) $\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = 4 R x ^ { 2 } y ^ { 2 }$

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 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) $x ^ { 2 } + y ^ { 2 } - 2 x ^ { 2 } y ^ { 2 } = 0$

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 $S _ { 1 } : x ^ { 2 } + y ^ { 2 } = 9$ and $S _ { 2 } : ( x - 2 ) ^ { 2 } + y ^ { 2 } = 1$.
Then the locus of center of a variable circle $S$ which touches $S _ { 1 }$ internally and $S _ { 2 }$ externally always passes through the points:
(1) $( 0 , \pm \sqrt { 3 } )$
(2) $\left( \frac { 1 } { 2 } , \pm \frac { \sqrt { 5 } } { 2 } \right)$
(3) $\left( 2 , \pm \frac { 3 } { 2 } \right)$
(4) $( 1 , \pm 2 )$

The locus of the midpoints of the chord of the circle, $x ^ { 2 } + y ^ { 2 } = 25$ which is tangent to the hyperbola, $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 16 } = 1$ is :
(1) $\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } - 16 x ^ { 2 } + 9 y ^ { 2 } = 0$
(2) $\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } - 9 x ^ { 2 } + 144 y ^ { 2 } = 0$
(3) $\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } - 9 x ^ { 2 } - 16 y ^ { 2 } = 0$
(4) $\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } - 9 x ^ { 2 } + 16 y ^ { 2 } = 0$

If two tangents drawn from a point $P$ to the parabola $y ^ { 2 } = 16 ( x - 3 )$ are at right angles, then the locus of point $P$ is: (1) $x + 4 = 0$ (2) $x + 2 = 0$ (3) $x + 3 = 0$ (4) $x + 1 = 0$

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 $\left( 0 , \frac { 3 } { 3 } \right)$

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 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) $\frac { 32 } { 3 }$

The locus of the middle points of the chords of the circle $C _ { 1 } : ( x - 4 ) ^ { 2 } + ( y - 5 ) ^ { 2 } = 4$ which subtend an angle $\theta _ { i }$ at the centre of the circle $C _ { i }$, is a circle of radius $r _ { i }$. If $\theta _ { 1 } = \frac { \pi } { 3 } , \theta _ { 3 } = \frac { 2 \pi } { 3 }$ and $r _ { 1 } ^ { 2 } = r _ { 2 } ^ { 2 } + r _ { 3 } ^ { 2 }$, then $\theta _ { 2 }$ is equal to
(1) $\frac { \pi } { 4 }$
(2) $\frac { 3 \pi } { 4 }$
(3) $\frac { \pi } { 6 }$
(4) $\frac { \pi } { 2 }$

Let $A$ be the point $(1, 2)$ and $B$ be any point on the curve $x ^ { 2 } + y ^ { 2 } = 16$. If the centre of the locus of the point $P$, which divides the line segment $AB$ in the ratio $3 : 2$ is the point $C( \alpha , \beta )$, then the length of the line segment $AC$ is
(1) $\frac { 3 \sqrt { 5 } } { 5 }$
(2) $\frac { 4 \sqrt { 5 } } { 5 }$
(3) $\frac { 2 \sqrt { 5 } } { 5 }$
(4) $\frac { 6 \sqrt { 5 } } { 5 }$

A line segment $AB$ of length $\lambda$ moves such that the points $A$ and $B$ remain on the periphery of a circle of radius $\lambda$. Then the locus of the point, that divides the line segment $AB$ in the ratio $2:3$, is a circle of radius
(1) $\frac{3}{5}\lambda$
(2) $\frac{2}{3}\lambda$
(3) $\frac{\sqrt{19}}{5}\lambda$
(4) $\frac{\sqrt{19}}{7}\lambda$

Let the locus of the mid points of the chords of circle $x^2 + (y-1)^2 = 1$ drawn from the origin intersect the line $x + y = 1$ at $P$ and $Q$. Then, the length of $PQ$ is:
(1) $\frac{1}{\sqrt{2}}$
(2) $\sqrt{2}$
(3) $\frac{1}{2}$
(4) 1

If the locus of the point, whose distances from the point $( 2,1 )$ and $( 1,3 )$ are in the ratio $5 : 4$, is $a x ^ { 2 } + b y ^ { 2 } + c x y + d x + e y + 170 = 0$, then the value of $a ^ { 2 } + 2 b + 3 c + 4 d + e$ is equal to:
(1) 37
(2) 437
(3) $- 27$
(4) 5

Consider the circle $C : x ^ { 2 } + y ^ { 2 } = 4$ and the parabola $P : y ^ { 2 } = 8 x$. If the set of all values of $\alpha$, for which three chords of the circle $C$ on three distinct lines passing through the point $( \alpha , 0 )$ are bisected by the parabola $P$ is the interval $( p , q )$, then $( 2 q - p ) ^ { 2 }$ is equal to $\_\_\_\_$