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 $\_\_\_\_$

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 $$M = \left\{ ( x , y ) \in \mathbb { R } \times \mathbb { R } : x ^ { 2 } + y ^ { 2 } \leq r ^ { 2 } \right\} ,$$ where $r > 0$. Consider the geometric progression $a _ { n } = \frac { 1 } { 2 ^ { n - 1 } } , n = 1,2,3 , \ldots$. Let $S _ { 0 } = 0$ and, for $n \geq 1$, let $S _ { n }$ denote the sum of the first $n$ terms of this progression. For $n \geq 1$, let $C _ { n }$ denote the circle with center ( $S _ { n - 1 } , 0$ ) and radius $a _ { n }$, and $D _ { n }$ denote the circle with center ( $S _ { n - 1 } , S _ { n - 1 }$ ) and radius $a _ { n }$. Consider $M$ with $r = \frac { 1025 } { 513 }$. Let $k$ be the number of all those circles $C _ { n }$ that are inside $M$. Let $l$ be the maximum possible number of circles among these $k$ circles such that no two circles intersect. Then
(A) $k + 2 l = 22$
(B) $2 k + l = 26$
(C) $2 k + 3 l = 34$
(D) $3 k + 2 l = 40$

Let $$M = \left\{ ( x , y ) \in \mathbb { R } \times \mathbb { R } : x ^ { 2 } + y ^ { 2 } \leq r ^ { 2 } \right\} ,$$ where $r > 0$. Consider the geometric progression $a _ { n } = \frac { 1 } { 2 ^ { n - 1 } } , n = 1,2,3 , \ldots$. Let $S _ { 0 } = 0$ and, for $n \geq 1$, let $S _ { n }$ denote the sum of the first $n$ terms of this progression. For $n \geq 1$, let $C _ { n }$ denote the circle with center ( $S _ { n - 1 } , 0$ ) and radius $a _ { n }$, and $D _ { n }$ denote the circle with center ( $S _ { n - 1 } , S _ { n - 1 }$ ) and radius $a _ { n }$. Consider $M$ with $r = \frac { \left( 2 ^ { 199 } - 1 \right) \sqrt { 2 } } { 2 ^ { 198 } }$. The number of all those circles $D _ { n }$ that are inside $M$ is
(A) 198
(B) 199
(C) 200
(D) 201

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 $\_\_\_\_$.

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 $\_\_\_\_$.

Let $G$ be a circle of radius $R > 0$. Let $G _ { 1 } , G _ { 2 } , \ldots , G _ { n }$ be $n$ circles of equal radius $r > 0$. Suppose each of the $n$ circles $G _ { 1 } , G _ { 2 } , \ldots , G _ { n }$ touches the circle $G$ externally. Also, for $i = 1,2 , \ldots , n - 1$, the circle $G _ { i }$ touches $G _ { i + 1 }$ externally, and $G _ { n }$ touches $G _ { 1 }$ externally. Then, which of the following statements is/are TRUE ?
(A) If $n = 4$, then $( \sqrt { 2 } - 1 ) r < R$
(B) If $n = 5$, then $r < R$
(C) If $n = 8$, then $( \sqrt { 2 } - 1 ) r < R$
(D) If $n = 12$, then $\sqrt { 2 } ( \sqrt { 3 } + 1 ) r > R$

Consider the parabola $y ^ { 2 } = 4 x$. Let $S$ be the focus of the parabola. A pair of tangents drawn to the parabola from the point $P = ( - 2,1 )$ meet the parabola at $P _ { 1 }$ and $P _ { 2 }$. Let $Q _ { 1 }$ and $Q _ { 2 }$ be points on the lines $S P _ { 1 }$ and $S P _ { 2 }$ respectively such that $P Q _ { 1 }$ is perpendicular to $S P _ { 1 }$ and $P Q _ { 2 }$ is perpendicular to $S P _ { 2 }$. Then, which of the following is/are TRUE?
(A) $\quad S Q _ { 1 } = 2$
(B) $\quad Q _ { 1 } Q _ { 2 } = \frac { 3 \sqrt { 10 } } { 5 }$
(C) $\quad P Q _ { 1 } = 3$
(D) $\quad S Q _ { 2 } = 1$

Consider the ellipse
$$\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 3 } = 1$$
Let $H ( \alpha , 0 ) , 0 < \alpha < 2$, be a point. A straight line drawn through $H$ parallel to the $y$-axis crosses the ellipse and its auxiliary circle at points $E$ and $F$ respectively, in the first quadrant. The tangent to the ellipse at the point $E$ intersects the positive $x$-axis at a point $G$. Suppose the straight line joining $F$ and the origin makes an angle $\phi$ with the positive $x$-axis.
List-I (I) If $\phi = \frac { \pi } { 4 }$, then the area of the triangle $F G H$ is (II) If $\phi = \frac { \pi } { 3 }$, then the area of the triangle $F G H$ is (III) If $\phi = \frac { \pi } { 6 }$, then the area of the triangle $F G H$ is (IV) If $\phi = \frac { \pi } { 12 }$, then the area of the triangle $F G H$ is
List-II (P) $\frac { ( \sqrt { 3 } - 1 ) ^ { 4 } } { 8 }$ (Q) 1 (R) $\frac { 3 } { 4 }$ (S) $\frac { 1 } { 2 \sqrt { 3 } }$ (T) $\frac { 3 \sqrt { 3 } } { 2 }$
The correct option is:
(A) (I) → (R); (II) → (S); (III) → (Q); (IV) → (P)
(B) (I) → (R); (II) → (T); (III) → (S); (IV) → (P)
(C) (I) → (Q); (II) → (T); (III) → (S); (IV) → (P)
(D) (I) → (Q); (II) → (S); (III) → (Q); (IV) → (P)

Let $T _ { 1 }$ and $T _ { 2 }$ be two distinct common tangents to the ellipse $E : \frac { x ^ { 2 } } { 6 } + \frac { y ^ { 2 } } { 3 } = 1$ and the parabola $P : y ^ { 2 } = 12 x$. Suppose that the tangent $T _ { 1 }$ touches $P$ and $E$ at the points $A _ { 1 }$ and $A _ { 2 }$, respectively and the tangent $T _ { 2 }$ touches $P$ and $E$ at the points $A _ { 4 }$ and $A _ { 3 }$, respectively. Then which of the following statements is(are) true?
(A) The area of the quadrilateral $A _ { 1 } A _ { 2 } A _ { 3 } A _ { 4 }$ is 35 square units
(B) The area of the quadrilateral $A _ { 1 } A _ { 2 } A _ { 3 } A _ { 4 }$ is 36 square units
(C) The tangents $T _ { 1 }$ and $T _ { 2 }$ meet the $x$-axis at the point $( - 3,0 )$
(D) The tangents $T _ { 1 }$ and $T _ { 2 }$ meet the $x$-axis at the point $( - 6,0 )$

Let $C _ { 1 }$ be the circle of radius 1 with center at the origin. Let $C _ { 2 }$ be the circle of radius $r$ with center at the point $A = ( 4,1 )$, where $1 < r < 3$. Two distinct common tangents $P Q$ and $S T$ of $C _ { 1 }$ and $C _ { 2 }$ are drawn. The tangent $P Q$ touches $C _ { 1 }$ at $P$ and $C _ { 2 }$ at $Q$. The tangent $S T$ touches $C _ { 1 }$ at $S$ and $C _ { 2 }$ at $T$. Mid points of the line segments $P Q$ and $S T$ are joined to form a line which meets the $x$-axis at a point $B$. If $A B = \sqrt { 5 }$, then the value of $r ^ { 2 }$ is

Let $A _ { 1 } , B _ { 1 } , C _ { 1 }$ be three points in the $xy$-plane. Suppose that the lines $A _ { 1 } C _ { 1 }$ and $B _ { 1 } C _ { 1 }$ are tangents to the curve $y ^ { 2 } = 8 x$ at $A _ { 1 }$ and $B _ { 1 }$, respectively. If $O = ( 0,0 )$ and $C _ { 1 } = ( - 4,0 )$, then which of the following statements is (are) TRUE?
(A) The length of the line segment $OA _ { 1 }$ is $4 \sqrt { 3 }$
(B) The length of the line segment $A _ { 1 } B _ { 1 }$ is 16
(C) The orthocenter of the triangle $A _ { 1 } B _ { 1 } C _ { 1 }$ is $( 0,0 )$
(D) The orthocenter of the triangle $A _ { 1 } B _ { 1 } C _ { 1 }$ is $( 1,0 )$

A normal with slope $\frac { 1 } { \sqrt { 6 } }$ is drawn from the point $( 0 , - \alpha )$ to the parabola $x ^ { 2 } = - 4 a y$, where $a > 0$. Let $L$ be the line passing through $( 0 , - \alpha )$ and parallel to the directrix of the parabola. Suppose that $L$ intersects the parabola at two points $A$ and $B$. Let $r$ denote the length of the latus rectum and $s$ denote the square of the length of the line segment $AB$. If $r : s = 1 : 16$, then the value of $24 a$ is $\_\_\_\_$ .

Let the straight line $y = 2 x$ touch a circle with center $( 0 , \alpha ) , \alpha > 0$, and radius $r$ at a point $A _ { 1 }$. Let $B _ { 1 }$ be the point on the circle such that the line segment $A _ { 1 } B _ { 1 }$ is a diameter of the circle. Let $\alpha + r = 5 + \sqrt { 5 }$.
Match each entry in List-I to the correct entry in List-II.
List-I
(P) $\alpha$ equals
(Q) $r$ equals
(R) $A _ { 1 }$ equals
(S) $B _ { 1 }$ equals
List-II
(1) $( - 2,4 )$
(2) $\sqrt { 5 }$
(3) $( - 2,6 )$
(4) 5
(5) $( 2,4 )$
The correct option is
(A) $( \mathrm { P } ) \rightarrow ( 4 )$, $( \mathrm { Q } ) \rightarrow ( 2 )$, $( \mathrm { R } ) \rightarrow ( 1 )$, $( \mathrm { S } ) \rightarrow ( 3 )$
(B) $( \mathrm { P } ) \rightarrow ( 2 )$, $( \mathrm { Q } ) \rightarrow ( 4 )$, $( \mathrm { R } ) \rightarrow ( 1 )$, $( \mathrm { S } ) \rightarrow ( 3 )$
(C) $( \mathrm { P } ) \rightarrow ( 4 )$, $( \mathrm { Q } ) \rightarrow ( 2 )$, $( \mathrm { R } ) \rightarrow ( 5 )$, $( \mathrm { S } ) \rightarrow ( 3 )$
(D) $( \mathrm { P } ) \rightarrow ( 2 )$, $( \mathrm { Q } ) \rightarrow ( 4 )$, $( \mathrm { R } ) \rightarrow ( 3 )$, $( \mathrm { S } ) \rightarrow ( 5 )$

Let $S$ denote the locus of the point of intersection of the pair of lines
$$\begin{gathered} 4 x - 3 y = 12 \alpha \\ 4 \alpha x + 3 \alpha y = 12 \end{gathered}$$
where $\alpha$ varies over the set of non-zero real numbers. Let $T$ be the tangent to $S$ passing through the points $( p , 0 )$ and $( 0 , q ) , q > 0$, and parallel to the line $4 x - \frac { 3 } { \sqrt { 2 } } y = 0$.
Then the value of $p q$ is

(A)

$- 6 \sqrt { 2 }$

(B)

$- 3 \sqrt { 2 }$

(C)

$- 9 \sqrt { 2 }$

(D)

$- 12 \sqrt { 2 }$

Let $\mathbb { R }$ denote the set of all real numbers. Let $z _ { 1 } = 1 + 2 i$ and $z _ { 2 } = 3 i$ be two complex numbers, where $i = \sqrt { - 1 }$. Let
$$S = \left\{ ( x , y ) \in \mathbb { R } \times \mathbb { R } : \left| x + i y - z _ { 1 } \right| = 2 \left| x + i y - z _ { 2 } \right| \right\}$$
Then which of the following statements is (are) TRUE?

(A)	$S$ is a circle with centre $\left( - \frac { 1 } { 3 } , \frac { 10 } { 3 } \right)$
(B)	$S$ is a circle with centre $\left( \frac { 1 } { 3 } , \frac { 8 } { 3 } \right)$
(C)	$S$ is a circle with radius $\frac { \sqrt { 2 } } { 3 }$
(D)	$S$ is a circle with radius $\frac { 2 \sqrt { 2 } } { 3 }$

Consider a family of circles which are passing through the point $( - 1,1 )$ and are tangent to $x -$ axis. If $( h , k )$ are the co-ordinates of the centre of the circles, then the set of values of $k$ is given by the interval
(1) $0 < \mathrm { k } < 1 / 2$
(2) $k \geq 1 / 2$
(3) $- 1 / 2 \leq k \leq 1 / 2$
(4) $k \leq 1 / 2$

The equation of a tangent to the parabola $y ^ { 2 } = 8 x$ is $y = x + 2$. The point on this line from which the other tangent to the parabola is perpendicular to the given tangent is
(1) $( - 1,1 )$
(2) $( 0,2 )$
(3) $( 2,4 )$
(4) $( - 2,0 )$

If ( $2,3,5$ ) is one end of a diameter of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 6 x - 12 y - 2 z + 20 = 0$, then the coordinates of the other end of the diameter are
(1) $( 4,9 , - 3 )$
(2) $( 4 , - 3,3 )$
(3) $( 4,3,5 )$
(4) $( 4,3 , - 3 )$