Tangents are drawn from the point $P ( 3,4 )$ to the ellipse $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 4 } = 1$ touching the ellipse at points A and B.
The equation of the locus of the point whose distances from the point $P$ and the line AB are equal, is
A) $9 x ^ { 2 } + y ^ { 2 } - 6 x y - 54 x - 62 y + 241 = 0$
B) $x ^ { 2 } + 9 y ^ { 2 } + 6 x y - 54 x + 62 y - 241 = 0$
C) $9 x ^ { 2 } + 9 y ^ { 2 } - 6 x y - 54 x - 62 y - 241 = 0$
D) $x ^ { 2 } + y ^ { 2 } - 2 x y + 27 x + 31 y - 120 = 0$

Let $a, r, s, t$ be nonzero real numbers. Let $P(at^2, 2at)$, $Q$, $R(ar^2, 2ar)$ and $S(as^2, 2as)$ be distinct points on the parabola $y^2 = 4ax$. Suppose that $PQ$ is the focal chord and lines $QR$ and $PK$ are parallel, where $K$ is the point $(2a, 0)$.
The value of $r$ is
(A) $-\frac{1}{t}$
(B) $\frac{t^2+1}{t}$
(C) $\frac{1}{t}$
(D) $\frac{t^2-1}{t}$

Let $RS$ be the diameter of the circle $x^2 + y^2 = 1$, where $S$ is the point $(1,0)$. Let $P$ be a variable point (other than $R$ and $S$) on the circle and tangents to the circle at $S$ and $P$ meet at the point $Q$. The normal to the circle at $P$ intersects a line drawn through $Q$ parallel to $RS$ at point $E$. Then the locus of $E$ passes through the point(s)
(A) $\left(\frac{1}{3}, \frac{1}{\sqrt{3}}\right)$
(B) $\left(\frac{1}{4}, \frac{1}{2}\right)$
(C) $\left(\frac{1}{3}, -\frac{1}{\sqrt{3}}\right)$
(D) $\left(\frac{1}{4}, -\frac{1}{2}\right)$

Let $T$ be the line passing through the points $P ( - 2,7 )$ and $Q ( 2 , - 5 )$. Let $F _ { 1 }$ be the set of all pairs of circles ( $S _ { 1 } , S _ { 2 }$ ) such that $T$ is tangent to $S _ { 1 }$ at $P$ and tangent to $S _ { 2 }$ at $Q$, and also such that $S _ { 1 }$ and $S _ { 2 }$ touch each other at a point, say, $M$. Let $E _ { 1 }$ be the set representing the locus of $M$ as the pair ( $S _ { 1 } , S _ { 2 }$ ) varies in $F _ { 1 }$. Let the set of all straight line segments joining a pair of distinct points of $E _ { 1 }$ and passing through the point $R ( 1,1 )$ be $F _ { 2 }$. Let $E _ { 2 }$ be the set of the mid-points of the line segments in the set $F _ { 2 }$. Then, which of the following statement(s) is (are) TRUE?
(A) The point $( - 2,7 )$ lies in $E _ { 1 }$
(B) The point $\left( \frac { 4 } { 5 } , \frac { 7 } { 5 } \right)$ does NOT lie in $E _ { 2 }$
(C) The point $\left( \frac { 1 } { 2 } , 1 \right)$ lies in $E _ { 2 }$
(D) The point $\left( 0 , \frac { 3 } { 2 } \right)$ does NOT lie in $E _ { 1 }$

Let $S$ be the circle in the $x y$-plane defined by the equation $x ^ { 2 } + y ^ { 2 } = 4$. Let $E _ { 1 } E _ { 2 }$ and $F _ { 1 } F _ { 2 }$ be the chords of $S$ passing through the point $P _ { 0 } ( 1,1 )$ and parallel to the $x$-axis and the $y$-axis, respectively. Let $G _ { 1 } G _ { 2 }$ be the chord of $S$ passing through $P _ { 0 }$ and having slope $- 1$. Let the tangents to $S$ at $E _ { 1 }$ and $E _ { 2 }$ meet at $E _ { 3 }$, the tangents to $S$ at $F _ { 1 }$ and $F _ { 2 }$ meet at $F _ { 3 }$, and the tangents to $S$ at $G _ { 1 }$ and $G _ { 2 }$ meet at $G _ { 3 }$. Then, the points $E _ { 3 } , F _ { 3 }$, and $G _ { 3 }$ lie on the curve
(A) $x + y = 4$
(B) $( x - 4 ) ^ { 2 } + ( y - 4 ) ^ { 2 } = 16$
(C) $( x - 4 ) ( y - 4 ) = 4$
(D) $x y = 4$

Let $S$ be the circle in the $x y$-plane defined by the equation $x ^ { 2 } + y ^ { 2 } = 4$. Let $P$ be a point on the circle $S$ with both coordinates being positive. Let the tangent to $S$ at $P$ intersect the coordinate axes at the points $M$ and $N$. Then, the mid-point of the line segment $M N$ must lie on the curve
(A) $( x + y ) ^ { 2 } = 3 x y$
(B) $x ^ { 2 / 3 } + y ^ { 2 / 3 } = 2 ^ { 4 / 3 }$
(C) $x ^ { 2 } + y ^ { 2 } = 2 x y$
(D) $x ^ { 2 } + y ^ { 2 } = x ^ { 2 } y ^ { 2 }$

The locus of the foot of perpendicular drawn from the centre of the ellipse $x ^ { 2 } + 3 y ^ { 2 } = 6$ on any tangent to it is
(1) $\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = 6 x ^ { 2 } + 2 y ^ { 2 }$
(2) $\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = 6 x ^ { 2 } - 2 y ^ { 2 }$
(3) $\left( x ^ { 2 } - y ^ { 2 } \right) ^ { 2 } = 6 x ^ { 2 } + 2 y ^ { 2 }$
(4) $\left( x ^ { 2 } - y ^ { 2 } \right) ^ { 2 } = 6 x ^ { 2 } - 2 y ^ { 2 }$

Let $a$ and $b$ be any two numbers satisfying $\frac { 1 } { a ^ { 2 } } + \frac { 1 } { b ^ { 2 } } = \frac { 1 } { 4 }$. Then, the foot of perpendicular from the origin on the variable line $\frac { x } { a } + \frac { y } { b } = 1$ lies on:
(1) A circle of radius $= 2$
(2) A hyperbola with each semi-axis $= \sqrt { 2 }$.
(3) A hyperbola with each semi-axis $= 2$
(4) A circle of radius $= \sqrt { 2 }$

Locus of the image of the point $(2, 3)$ in the line $(2x - 3y + 4) + k(x - 2y + 3) = 0$, $k \in \mathbb{R}$, is a:
(1) straight line parallel to $x$-axis
(2) straight line parallel to $y$-axis
(3) circle of radius $\sqrt{2}$
(4) circle of radius $\sqrt{3}$

Let $O$ be the vertex and $Q$ be any point on the parabola, $x ^ { 2 } = 8 y$. If the point $P$ divides the line segment $OQ$ internally in the ratio $1 : 3$, then the locus of $P$ is
(1) $x ^ { 2 } = 2 y$
(2) $x ^ { 2 } = y$
(3) $y ^ { 2 } = x$
(4) $y ^ { 2 } = 2 x$

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) $\frac { \pi } { 2 }$

The ordinates of the points $P$ and $Q$ on the parabola with focus $( 3,0 )$ and directrix $x = - 3$ are in the ratio 3:1. If $R ( \alpha , \beta )$ is the point of intersection of the tangents to the parabola at $P$ and $Q$, then $\frac { \beta ^ { 2 } } { \alpha }$ is equal to