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) $( - 4,0 )$

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) $x ^ { 2 } + y ^ { 2 } - 6 x - 7 y + 9 = 0$

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

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

If the normals of the parabola $y ^ { 2 } = 4 x$ drawn at the end points of its latus rectum are tangents to the circle $( x - 3 ) ^ { 2 } + ( y + 2 ) ^ { 2 } = r ^ { 2 }$, then the value of $r ^ { 2 }$ is

Let $E _ { 1 }$ and $E _ { 2 }$ be two ellipses whose centers are at the origin. The major axes of $E _ { 1 }$ and $E _ { 2 }$ lie along the $x$-axis and the $y$-axis, respectively. Let $S$ be the circle $x ^ { 2 } + ( y - 1 ) ^ { 2 } = 2$. The straight line $x + y = 3$ touches the curves $S , E _ { 1 }$ and $E _ { 2 }$ at $P , Q$ and $R$, respectively. Suppose that $P Q = P R = \frac { 2 \sqrt { 2 } } { 3 }$. If $e _ { 1 }$ and $e _ { 2 }$ are the eccentricities of $E _ { 1 }$ and $E _ { 2 }$, respectively, then the correct expression(s) is(are)
(A) $e _ { 1 } ^ { 2 } + e _ { 2 } ^ { 2 } = \frac { 43 } { 40 }$
(B) $\quad e _ { 1 } e _ { 2 } = \frac { \sqrt { 7 } } { 2 \sqrt { 10 } }$
(C) $\left| e _ { 1 } ^ { 2 } - e _ { 2 } ^ { 2 } \right| = \frac { 5 } { 8 }$
(D) $e _ { 1 } e _ { 2 } = \frac { \sqrt { 3 } } { 4 }$

Let $P$ and $Q$ be distinct points on the parabola $y ^ { 2 } = 2 x$ such that a circle with $P Q$ as diameter passes through the vertex $O$ of the parabola. If $P$ lies in the first quadrant and the area of the triangle $\triangle O P Q$ is $3 \sqrt { 2 }$, then which of the following is (are) the coordinates of $P$?
(A) $( 4,2 \sqrt { 2 } )$
(B) $( 9,3 \sqrt { 2 } )$
(C) $\left( \frac { 1 } { 4 } , \frac { 1 } { \sqrt { 2 } } \right)$
(D) $( 1 , \sqrt { 2 } )$

The circle $C_1: x^2 + y^2 = 3$, with centre at $O$, intersects the parabola $x^2 = 2y$ at the point $P$ in the first quadrant. Let the tangent to the circle $C_1$ at $P$ touches other two circles $C_2$ and $C_3$ at $R_2$ and $R_3$, respectively. Suppose $C_2$ and $C_3$ have equal radii $2\sqrt{3}$ and centres $Q_2$ and $Q_3$, respectively. If $Q_2$ and $Q_3$ lie on the $y$-axis, then
(A) $Q_2 Q_3 = 12$
(B) $R_2 R_3 = 4\sqrt{6}$
(C) area of the triangle $OR_2R_3$ is $6\sqrt{2}$
(D) area of the triangle $PQ_2Q_3$ is $4\sqrt{2}$

Let $P$ be the point on the parabola $y ^ { 2 } = 4 x$ which is at the shortest distance from the center $S$ of the circle $x ^ { 2 } + y ^ { 2 } - 4 x - 16 y + 64 = 0$. Let $Q$ be the point on the circle dividing the line segment $S P$ internally. Then
(A) $S P = 2 \sqrt { 5 }$
(B) $S Q : Q P = ( \sqrt { 5 } + 1 ) : 2$
(C) the $x$-intercept of the normal to the parabola at $P$ is 6
(D) the slope of the tangent to the circle at $Q$ is $\frac { 1 } { 2 }$

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) $\left(\frac{1}{4}, -\frac{1}{2}\right)$

If a chord, which is not a tangent, of the parabola $y^2 = 16x$ has the equation $2x + y = p$, and midpoint $(h, k)$, then which of the following is(are) possible value(s) of $p$, $h$ and $k$?
[A] $p = -2, h = 2, k = -4$
[B] $p = -1, h = 1, k = -3$
[C] $p = 2, h = 3, k = -4$
[D] $p = 5, h = 4, k = -3$

For how many values of $p$, the circle $x^2 + y^2 + 2x + 4y - p = 0$ and the coordinate axes have exactly three common points?

Columns 1, 2 and 3 contain conics, equations of tangents to the conics and points of contact, respectively.

	Column 1	Column 2	Column 3
	(I) $x^2 + y^2 = a^2$	(i) $my = m^2x + a$	(P) $\left(\frac{a}{m^2}, \frac{2a}{m}\right)$
(II)	$x^2 + a^2y^2 = a^2$	(ii) $y = mx + a\sqrt{m^2+1}$	(Q) $\left(\frac{-ma}{\sqrt{m^2+1}}, \frac{a}{\sqrt{m^2+1}}\right)$
(III)	$y^2 = 4ax$	(iii) $y = mx + \sqrt{a^2m^2 - 1}$	(R) $\left(\frac{-a^2m}{\sqrt{a^2m^2+1}}, \frac{1}{\sqrt{a^2m^2+1}}\right)$
(IV)	$x^2 - a^2y^2 = a^2$	(iv) $y = mx + \sqrt{a^2m^2+1}$	(S) $\left(\frac{-a^2m}{\sqrt{a^2m^2-1}}, \frac{-1}{\sqrt{a^2m^2-1}}\right)$

For $a = \sqrt{2}$, if a tangent is drawn to a suitable conic (Column 1) at the point of contact $(-1, 1)$, then which of the following options is the only CORRECT combination for obtaining its equation?
[A] (I) (i) (P)
[B] (I) (ii) (Q)
[C] (II) (ii) (Q)
[D] (III) (i) (P)

Columns 1, 2 and 3 contain conics, equations of tangents to the conics and points of contact, respectively.

	Column 1	Column 2	Column 3
	(I) $x^2 + y^2 = a^2$	(i) $my = m^2x + a$	(P) $\left(\frac{a}{m^2}, \frac{2a}{m}\right)$
(II)	$x^2 + a^2y^2 = a^2$	(ii) $y = mx + a\sqrt{m^2+1}$	(Q) $\left(\frac{-ma}{\sqrt{m^2+1}}, \frac{a}{\sqrt{m^2+1}}\right)$
(III)	$y^2 = 4ax$	(iii) $y = mx + \sqrt{a^2m^2 - 1}$	(R) $\left(\frac{-a^2m}{\sqrt{a^2m^2+1}}, \frac{1}{\sqrt{a^2m^2+1}}\right)$
(IV)	$x^2 - a^2y^2 = a^2$	(iv) $y = mx + \sqrt{a^2m^2+1}$	(S) $\left(\frac{-a^2m}{\sqrt{a^2m^2-1}}, \frac{-1}{\sqrt{a^2m^2-1}}\right)$

If a tangent to a suitable conic (Column 1) is found to be $y = x + 8$ and its point of contact is $(8, 16)$, then which of the following options is the only CORRECT combination?
[A] (I) (ii) (Q)
[B] (II) (iv) (R)
[C] (III) (i) (P)
[D] (III) (ii) (Q)

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 $E _ { 1 }$

Consider two straight lines, each of which is tangent to both the circle $x ^ { 2 } + y ^ { 2 } = \frac { 1 } { 2 }$ and the parabola $y ^ { 2 } = 4 x$. Let these lines intersect at the point $Q$. Consider the ellipse whose center is at the origin $O ( 0,0 )$ and whose semi-major axis is $O Q$. If the length of the minor axis of this ellipse is $\sqrt { 2 }$, then which of the following statement(s) is (are) TRUE?
(A) For the ellipse, the eccentricity is $\frac { 1 } { \sqrt { 2 } }$ and the length of the latus rectum is 1
(B) For the ellipse, the eccentricity is $\frac { 1 } { 2 }$ and the length of the latus rectum is $\frac { 1 } { 2 }$
(C) The area of the region bounded by the ellipse between the lines $x = \frac { 1 } { \sqrt { 2 } }$ and $x = 1$ is $\frac { 1 } { 4 \sqrt { 2 } } ( \pi - 2 )$
(D) The area of the region bounded by the ellipse between the lines $x = \frac { 1 } { \sqrt { 2 } }$ and $x = 1$ is
$$\frac { 1 } { 16 } ( \pi - 2 )$$

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

A line $y = m x + 1$ intersects the circle $( x - 3 ) ^ { 2 } + ( y + 2 ) ^ { 2 } = 25$ at the points $P$ and $Q$. If the midpoint of the line segment $P Q$ has $x$-coordinate $- \frac { 3 } { 5 }$, then which one of the following options is correct?
(A) $\quad - 3 \leq m < - 1$
(B) $2 \leq m < 4$
(C) $4 \leq m < 6$
(D) $6 \leq m < 8$

Define the collections $\left\{ E _ { 1 } , E _ { 2 } , E _ { 3 } , \ldots \right\}$ of ellipses and $\left\{ R _ { 1 } , R _ { 2 } , R _ { 3 } , \ldots \right\}$ of rectangles as follows: $E _ { 1 } : \frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 4 } = 1 ;$ $R _ { 1 }$ : rectangle of largest area, with sides parallel to the axes, inscribed in $E _ { 1 }$; $E _ { n }$ : ellipse $\frac { x ^ { 2 } } { a _ { n } ^ { 2 } } + \frac { y ^ { 2 } } { b _ { n } ^ { 2 } } = 1$ of largest area inscribed in $R _ { n - 1 } , n > 1$; $R _ { n }$ : rectangle of largest area, with sides parallel to the axes, inscribed in $E _ { n } , n > 1$. Then which of the following options is/are correct?
(A) The eccentricities of $E _ { 18 }$ and $E _ { 19 }$ are NOT equal
(B) $\quad \sum _ { n = 1 } ^ { N } \left( \right.$ area of $\left. R _ { n } \right) < 24$, for each positive integer $N$
(C) The length of latus rectum of $E _ { 9 }$ is $\frac { 1 } { 6 }$
(D) The distance of a focus from the centre in $E _ { 9 }$ is $\frac { \sqrt { 5 } } { 32 }$