A circle $C$ of radius 1 is inscribed in an equilateral triangle $P Q R$. The points of contact of $C$ with the sides $P Q , Q R , R P$ are $D , E , F$, respectively. The line $P Q$ is given by the equation $\sqrt { 3 } x + y - 6 = 0$ and the point $D$ is $\left( \frac { 3 \sqrt { 3 } } { 2 } , \frac { 3 } { 2 } \right)$. Further, it is given that the origin and the centre of $C$ are on the same side of the line $P Q$.
Equations of the sides $Q R , R P$ are
(A) $y = \frac { 2 } { \sqrt { 3 } } x + 1 , y = - \frac { 2 } { \sqrt { 3 } } x - 1$
(B) $y = \frac { 1 } { \sqrt { 3 } } x , y = 0$
(C) $y = \frac { \sqrt { 3 } } { 2 } x + 1 , y = - \frac { \sqrt { 3 } } { 2 } x - 1$
(D) $y = \sqrt { 3 } x , y = 0$

Tangents drawn from the point $P ( 1,8 )$ to the circle
$$x ^ { 2 } + y ^ { 2 } - 6 x - 4 y - 11 = 0$$
touch the circle at the points $A$ and $B$. The equation of the circumcircle of the triangle $P A B$ is
(A) $x ^ { 2 } + y ^ { 2 } + 4 x - 6 y + 19 = 0$
(B) $x ^ { 2 } + y ^ { 2 } - 4 x - 10 y + 19 = 0$
(C) $x ^ { 2 } + y ^ { 2 } - 2 x + 6 y - 29 = 0$
(D) $x ^ { 2 } + y ^ { 2 } - 6 x - 4 y + 19 = 0$

The normal at a point $P$ on the ellipse $x^{2}+4y^{2}=16$ meets the $x$-axis at $Q$. If $M$ is the mid point of the line segment $PQ$, then the locus of $M$ intersects the latus rectums of the given ellipse at the points
(A) $\left(\pm\frac{3\sqrt{5}}{2},\pm\frac{2}{7}\right)$
(B) $\left(\pm\frac{3\sqrt{5}}{2},\pm\frac{\sqrt{19}}{4}\right)$
(C) $\left(\pm2\sqrt{3},\pm\frac{1}{7}\right)$
(D) $\left(\pm2\sqrt{3},\pm\frac{4\sqrt{3}}{7}\right)$

The line passing through the extremity $A$ of the major axis and extremity $B$ of the minor axis of the ellipse
$$x ^ { 2 } + 9 y ^ { 2 } = 9$$
meets its auxiliary circle at the point $M$. Then the area of the triangle with vertices at $A , M$ and the origin $O$ is
(A) $\frac { 31 } { 10 }$
(B) $\frac { 29 } { 10 }$
(C) $\frac { 21 } { 10 }$
(D) $\frac { 27 } { 10 }$

An ellipse intersects the hyperbola $2x^{2}-2y^{2}=1$ orthogonally. The eccentricity of the ellipse is reciprocal of that of the hyperbola. If the axes of the ellipse are along the coordinate axes, then
(A) Equation of ellipse is $x^{2}+2y^{2}=2$
(B) The foci of ellipse are $(\pm1,0)$
(C) Equation of ellipse is $x^{2}+2y^{2}=4$
(D) The foci of ellipse are $(\pm\sqrt{2},0)$

The centres of two circles $C_{1}$ and $C_{2}$ each of unit radius are at a distance of 6 units from each other. Let $P$ be the mid point of the line segment joining the centres of $C_{1}$ and $C_{2}$ and $C$ be a circle touching circles $C_{1}$ and $C_{2}$ externally. If a common tangent to $C_{1}$ and $C$ passing through $P$ is also a common tangent to $C_{2}$ and $C$, then the radius of the circle $C$ is

Two parallel chords of a circle of radius 2 are at a distance $\sqrt { 3 } + 1$ apart. If the chords subtend at the center, angles of $\frac { \pi } { k }$ and $\frac { 2 \pi } { k }$, where $k > 0$, then the value of $[ k ]$ is [Note : [k] denotes the largest integer less than or equal to k]

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 coordinates of $A$ and $B$ are
A) $( 3,0 )$ and $( 0,2 )$
B) $\left( - \frac { 8 } { 5 } , \frac { 2 \sqrt { 161 } } { 15 } \right)$ and $\left( - \frac { 9 } { 5 } , \frac { 8 } { 5 } \right)$
C) $\left( - \frac { 8 } { 5 } , \frac { 2 \sqrt { 161 } } { 15 } \right)$ and $( 0,2 )$
D) $(3, 0)$ and $\left( - \frac { 9 } { 5 } , \frac { 8 } { 5 } \right)$

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 orthocenter of the triangle $P A B$ is
A) $\left( 5 , \frac { 8 } { 7 } \right)$
B) $\left( \frac { 7 } { 5 } , \frac { 25 } { 8 } \right)$
C) $\left( \frac { 11 } { 5 } , \frac { 8 } { 5 } \right)$
D) $\left( \frac { 8 } { 25 } , \frac { 7 } { 5 } \right)$

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 and B be two distinct points on the parabola $\mathrm { y } ^ { 2 } = 4 \mathrm { x }$. If the axis of the parabola touches a circle of radius $r$ having $A B$ as its diameter, then the slope of the line joining A and B can be
A) $- \frac { 1 } { r }$
B) $\frac { 1 } { r }$
C) $\frac { 2 } { r }$
D) $- \frac { 2 } { \mathrm { r } }$

The circle $x ^ { 2 } + y ^ { 2 } - 8 x = 0$ and hyperbola $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 4 } = 1$ intersect at the points $A$ and $B$.
Equation of a common tangent with positive slope to the circle as well as to the hyperbola is
A) $2 x - \sqrt { 5 } y - 20 = 0$
B) $2 x - \sqrt { 5 } y + 4 = 0$
C) $3 x - 4 y + 8 = 0$
D) $4 x - 3 y + 4 = 0$

The circle $x ^ { 2 } + y ^ { 2 } - 8 x = 0$ and hyperbola $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 4 } = 1$ intersect at the points $A$ and $B$.
Equation of the circle with AB as its diameter is
A) $x ^ { 2 } + y ^ { 2 } - 12 x + 24 = 0$
B) $x ^ { 2 } + y ^ { 2 } + 12 x + 24 = 0$
C) $x ^ { 2 } + y ^ { 2 } + 24 x - 12 = 0$
D) $x ^ { 2 } + y ^ { 2 } - 24 x - 12 = 0$

The circle passing through the point $( - 1,0 )$ and touching the $y$-axis at $( 0,2 )$ also passes through the point
(A) $\left( - \frac { 3 } { 2 } , 0 \right)$
(B) $\left( - \frac { 5 } { 2 } , 2 \right)$
(C) $\left( - \frac { 3 } { 2 } , \frac { 5 } { 2 } \right)$
(D) $( - 4,0 )$

Let $( x , y )$ be any point on the parabola $y ^ { 2 } = 4 x$. Let $P$ be the point that divides the line segment from $( 0,0 )$ to $( x , y )$ in the ratio $1 : 3$. Then the locus of $P$ is
(A) $x ^ { 2 } = y$
(B) $y ^ { 2 } = 2 x$
(C) $y ^ { 2 } = x$
(D) $x ^ { 2 } = 2 y$

Circle(s) touching $x$-axis at a distance 3 from the origin and having an intercept of length $2 \sqrt { 7 }$ on $y$-axis is (are)
(A) $x ^ { 2 } + y ^ { 2 } - 6 x + 8 y + 9 = 0$
(B) $x ^ { 2 } + y ^ { 2 } - 6 x + 7 y + 9 = 0$
(C) $x ^ { 2 } + y ^ { 2 } - 6 x - 8 y + 9 = 0$
(D) $x ^ { 2 } + y ^ { 2 } - 6 x - 7 y + 9 = 0$

Let $P Q$ be a focal chord of the parabola $y ^ { 2 } = 4 a x$. The tangents to the parabola at $P$ and $Q$ meet at a point lying on the line $y = 2 x + a , a > 0$.
Length of chord $P Q$ is
(A) $7 a$
(B) $5 a$
(C) $2 a$
(D) $3 a$

Let $P Q$ be a focal chord of the parabola $y ^ { 2 } = 4 a x$. The tangents to the parabola at $P$ and $Q$ meet at a point lying on the line $y = 2 x + a , a > 0$.
If chord $P Q$ subtends an angle $\theta$ at the vertex of $y ^ { 2 } = 4 a x$, then $\tan \theta =$
(A) $\frac { 2 } { 3 } \sqrt { 7 }$
(B) $\frac { - 2 } { 3 } \sqrt { 7 }$
(C) $\frac { 2 } { 3 } \sqrt { 5 }$
(D) $\frac { - 2 } { 3 } \sqrt { 5 }$

A vertical line passing through the point $( h , 0 )$ intersects the ellipse $\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 3 } = 1$ at the points $P$ and $Q$. Let the tangents to the ellipse at $P$ and $Q$ meet at the point $R$. If $\Delta ( h ) =$ area of the triangle $P Q R$, $\Delta _ { 1 } = \max _ { 1/2 \leq h \leq 1 } \Delta ( h )$ and $\Delta _ { 2 } = \min _ { 1/2 \leq h \leq 1 } \Delta ( h )$, then $\frac { 8 } { \sqrt { 5 } } \Delta _ { 1 } - 8 \Delta _ { 2 } =$

The common tangents to the circle $x^2 + y^2 = 2$ and the parabola $y^2 = 8x$ touch the circle at the points $P, Q$ and the parabola at the points $R, S$. Then the area of the quadrilateral $PQRS$ is
(A) 3
(B) 6
(C) 9
(D) 15

A circle $S$ passes through the point $(0,1)$ and is orthogonal to the circles $(x-1)^2 + y^2 = 16$ and $x^2 + y^2 = 1$. Then
(A) radius of $S$ is 8
(B) radius of $S$ is 7
(C) centre of $S$ is $(-7, 1)$
(D) centre of $S$ is $(-8, 1)$

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) $\frac{t^2-1}{t}$

If $st = 1$, then the tangent at $P$ and the normal at $S$ to the parabola meet at a point whose ordinate is
(A) $\frac{(t^2+1)^2}{2t^3}$
(B) $\frac{a(t^2+1)^2}{2t^3}$
(C) $\frac{a(t^2+1)^2}{t^3}$
(D) $\frac{a(t^2+2)^2}{t^3}$

Let the curve $C$ be the mirror image of the parabola $y ^ { 2 } = 4 x$ with respect to the line $x + y + 4 = 0$. If $A$ and $B$ are the points of intersection of $C$ with the line $y = - 5$, then the distance between $A$ and $B$ is

Suppose that the foci of the ellipse $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 5 } = 1$ are $\left( f _ { 1 } , 0 \right)$ and $\left( f _ { 2 } , 0 \right)$ where $f _ { 1 } > 0$ and $f _ { 2 } < 0$. Let $P _ { 1 }$ and $P _ { 2 }$ be two parabolas with a common vertex at ( 0,0 ) and with foci at ( $f _ { 1 } , 0$ ) and ( $2 f _ { 2 } , 0$ ), respectively. Let $T _ { 1 }$ be a tangent to $P _ { 1 }$ which passes through ( $2 f _ { 2 } , 0$ ) and $T _ { 2 }$ be a tangent to $P _ { 2 }$ which passes through $\left( f _ { 1 } , 0 \right)$. If $m _ { 1 }$ is the slope of $T _ { 1 }$ and $m _ { 2 }$ is the slope of $T _ { 2 }$, then the value of $\left( \frac { 1 } { m _ { 1 } ^ { 2 } } + m _ { 2 } ^ { 2 } \right)$ is