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?

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) $6 \leq m < 8$

Let the point $B$ be the reflection of the point $A ( 2,3 )$ with respect to the line $8 x - 6 y - 23 = 0$. Let $\Gamma _ { A }$ and $\Gamma _ { B }$ be circles of radii 2 and 1 with centres $A$ and $B$ respectively. Let $T$ be a common tangent to the circles $\Gamma _ { A }$ and $\Gamma _ { B }$ such that both the circles are on the same side of $T$. If $C$ is the point of intersection of $T$ and the line passing through $A$ and $B$, then the length of the line segment $A C$ is $\_\_\_\_$

Let the circles $C_1 : x^2 + y^2 = 9$ and $C_2 : (x-3)^2 + (y-4)^2 = 16$, intersect at the points $X$ and $Y$. Suppose that another circle $C_3 : (x-h)^2 + (y-k)^2 = r^2$ satisfies the following conditions: (i) centre of $C_3$ is collinear with the centres of $C_1$ and $C_2$, (ii) $C_1$ and $C_2$ both lie inside $C_3$, and (iii) $C_3$ touches $C_1$ at $M$ and $C_2$ at $N$.
Let the line through $X$ and $Y$ intersect $C_3$ at $Z$ and $W$, and let a common tangent of $C_1$ and $C_3$ be a tangent to the parabola $x^2 = 8\alpha y$.
List-I: (I) $2h + k$ (II) $\frac{\text{Length of } ZW}{\text{Length of } XY}$ (III) $\frac{\text{Area of triangle } MZN}{\text{Area of triangle } ZMW}$ (IV) $\alpha$
List-II: (P) $6$ (Q) $\sqrt{6}$ (R) $\frac{5}{4}$ (S) $\frac{21}{5}$ (T) $2\sqrt{6}$ (U) $\frac{10}{3}$
Which of the following is the only CORRECT combination?
(A) (I), (S)
(B) (I), (U)
(C) (II), (Q)
(D) (II), (T)

Let the circles $C_1 : x^2 + y^2 = 9$ and $C_2 : (x-3)^2 + (y-4)^2 = 16$, intersect at the points $X$ and $Y$. Suppose that another circle $C_3 : (x-h)^2 + (y-k)^2 = r^2$ satisfies the following conditions: (i) centre of $C_3$ is collinear with the centres of $C_1$ and $C_2$, (ii) $C_1$ and $C_2$ both lie inside $C_3$, and (iii) $C_3$ touches $C_1$ at $M$ and $C_2$ at $N$.
Let the line through $X$ and $Y$ intersect $C_3$ at $Z$ and $W$, and let a common tangent of $C_1$ and $C_3$ be a tangent to the parabola $x^2 = 8\alpha y$.
List-I: (I) $2h + k$ (II) $\frac{\text{Length of } ZW}{\text{Length of } XY}$ (III) $\frac{\text{Area of triangle } MZN}{\text{Area of triangle } ZMW}$ (IV) $\alpha$
List-II: (P) $6$ (Q) $\sqrt{6}$ (R) $\frac{5}{4}$ (S) $\frac{21}{5}$ (T) $2\sqrt{6}$ (U) $\frac{10}{3}$
Which of the following is the only INCORRECT combination?
(A) (I), (P)
(B) (IV), (U)
(C) (III), (R)
(D) (IV), (S)

Let $O$ be the centre of the circle $x^{2} + y^{2} = r^{2}$, where $r > \frac{\sqrt{5}}{2}$. Suppose $PQ$ is a chord of this circle and the equation of the line passing through $P$ and $Q$ is $2x + 4y = 5$. If the centre of the circumcircle of the triangle $OPQ$ lies on the line $x + 2y = 4$, then the value of $r$ is $\_\_\_\_$

Let $a , b$ and $\lambda$ be positive real numbers. Suppose $P$ is an end point of the latus rectum of the parabola $y ^ { 2 } = 4 \lambda x$, and suppose the ellipse $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ passes through the point $P$. If the tangents to the parabola and the ellipse at the point $P$ are perpendicular to each other, then the eccentricity of the ellipse is
(A) $\frac { 1 } { \sqrt { 2 } }$
(B) $\frac { 1 } { 2 }$
(C) $\frac { 1 } { 3 }$
(D) $\frac { 2 } { 5 }$

Let $a$ and $b$ be positive real numbers such that $a > 1$ and $b < a$. Let $P$ be a point in the first quadrant that lies on the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$. Suppose the tangent to the hyperbola at $P$ passes through the point $(1, 0)$, and suppose the normal to the hyperbola at $P$ cuts off equal intercepts on the coordinate axes. Let $\Delta$ denote the area of the triangle formed by the tangent at $P$, the normal at $P$ and the $x$-axis. If $e$ denotes the eccentricity of the hyperbola, then which of the following statements is/are TRUE?
(A) $1 < e < \sqrt{2}$
(B) $\sqrt{2} < e < 2$
(C) $\Delta = a^{4}$
(D) $\Delta = b^{4}$

Consider a triangle $\Delta$ whose two sides lie on the x-axis and the line $x + y + 1 = 0$. If the orthocenter of $\Delta$ is $(1,1)$, then the equation of the circle passing through the vertices of the triangle $\Delta$ is
(A) $x^2 + y^2 - 3x + y = 0$
(B) $x^2 + y^2 + x + 3y = 0$
(C) $x^2 + y^2 + 2y - 1 = 0$
(D) $x^2 + y^2 + x + y = 0$

Let $E$ denote the parabola $y ^ { 2 } = 8 x$. Let $P = ( - 2,4 )$, and let $Q$ and $Q ^ { \prime }$ be two distinct points on $E$ such that the lines $P Q$ and $P Q ^ { \prime }$ are tangents to $E$. Let $F$ be the focus of $E$. Then which of the following statements is (are) TRUE ?
(A) The triangle $P F Q$ is a right-angled triangle
(B) The triangle $Q P Q ^ { \prime }$ is a right-angled triangle
(C) The distance between $P$ and $F$ is $5 \sqrt { 2 }$
(D) $F$ lies on the line joining $Q$ and $Q ^ { \prime }$

Consider the region $R = \left\{ ( x , y ) \in \mathbb { R } \times \mathbb { R } : x \geq 0 \right.$ and $\left. y ^ { 2 } \leq 4 - x \right\}$. Let $\mathcal { F }$ be the family of all circles that are contained in $R$ and have centers on the $x$-axis. Let $C$ be the circle that has largest radius among the circles in $\mathcal { F }$. Let $( \alpha , \beta )$ be a point where the circle $C$ meets the curve $y ^ { 2 } = 4 - x$. The radius of the circle $C$ is $\_\_\_\_$.

Consider the region $R = \left\{ ( x , y ) \in \mathbb { R } \times \mathbb { R } : x \geq 0 \right.$ and $\left. y ^ { 2 } \leq 4 - x \right\}$. Let $\mathcal { F }$ be the family of all circles that are contained in $R$ and have centers on the $x$-axis. Let $C$ be the circle that has largest radius among the circles in $\mathcal { F }$. Let $( \alpha , \beta )$ be a point where the circle $C$ meets the curve $y ^ { 2 } = 4 - x$. The value of $\alpha$ is $\_\_\_\_$.

Let $E$ be the ellipse $\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 9 } = 1$. For any three distinct points $P , Q$ and $Q ^ { \prime }$ on $E$, let $M ( P , Q )$ be the mid-point of the line segment joining $P$ and $Q$, and $M \left( P , Q ^ { \prime } \right)$ be the mid-point of the line segment joining $P$ and $Q ^ { \prime }$. Then the maximum possible value of the distance between $M ( P , Q )$ and $M \left( P , Q ^ { \prime } \right)$, as $P , Q$ and $Q ^ { \prime }$ vary on $E$, is $\_\_\_\_$.

Consider the hyperbola
$$\frac { x ^ { 2 } } { 100 } - \frac { y ^ { 2 } } { 64 } = 1$$
with foci at $S$ and $S _ { 1 }$, where $S$ lies on the positive $x$-axis. Let $P$ be a point on the hyperbola, in the first quadrant. Let $\angle S P S _ { 1 } = \alpha$, with $\alpha < \frac { \pi } { 2 }$. The straight line passing through the point $S$ and having the same slope as that of the tangent at $P$ to the hyperbola, intersects the straight line $S _ { 1 } P$ at $P _ { 1 }$. Let $\delta$ be the distance of $P$ from the straight line $S P _ { 1 }$, and $\beta = S _ { 1 } P$. Then the greatest integer less than or equal to $\frac { \beta \delta } { 9 } \sin \frac { \alpha } { 2 }$ is $\_\_\_\_$ .

Let $A B C$ be the triangle with $A B = 1 , A C = 3$ and $\angle B A C = \frac { \pi } { 2 }$. If a circle of radius $r > 0$ touches the sides $A B , A C$ and also touches internally the circumcircle of the triangle $A B C$, then the value of $r$ is $\_\_\_\_$.