The angle subtended at the origin by the common chord of the circles $x^2 + y^2 - 6x - 6y = 0$ and $x^2 + y^2 = 36$ is
(A) $\pi/2$
(B) $\pi/4$
(C) $\pi/3$
(D) $2\pi/3$

Consider a circle with centre $O$. Two chords $A B$ and $C D$ extended intersect at a point $P$ outside the circle. If $\angle A O C = 43 ^ { \circ }$ and $\angle B P D = 18 ^ { \circ }$, then the value of $\angle B O D$ is
(a) $36 ^ { \circ }$.
(B) $29 ^ { \circ }$.
(C) $7 ^ { \circ }$.
(D) $25 ^ { \circ }$.

16. If two distinct chords, drawn from the point ( $p , q$ ) on the circle $x 2 + y 2 = p x + q y$ (where $p q { } ^ { 1 } 0$ ) are bisected by the $x$-axis, then :
(A) $\mathrm { p } 2 = \mathrm { q } 2$
(B) $p 2 = 8 q 2$
(C) $p 2 < 8 q 2$
(D) $p 2 > 8 q 2$

26. Let L1 be a straight line passing through the origin and L2 be the straight line $x + y =$ 1. If the intercepts made by the circle $x 2 + y 2 - x + 3 y = 0$ on L1 and L2 are equal, then which of the following equations can represent L1?
(A) $x + y = 0$
(B) $x - y = 0$
(B) $x + 7 y = 0$
(D) $x - 7 y = 0$

A straight line through the vertex $P$ of a triangle $P Q R$ intersects the side $Q R$ at the point $S$ and the circumcircle of the triangle $P Q R$ at the point $T$. If $S$ is not the centre of the circumcircle, then
(A) $\frac { 1 } { P S } + \frac { 1 } { S T } < \frac { 2 } { \sqrt { Q S \times S R } }$
(B) $\frac { 1 } { P S } + \frac { 1 } { S T } > \frac { 2 } { \sqrt { Q S \times S R } }$
(C) $\frac { 1 } { P S } + \frac { 1 } { S T } < \frac { 4 } { Q R }$
(D) $\frac { 1 } { P S } + \frac { 1 } { S T } > \frac { 4 } { Q R }$

Consider
$$\begin{aligned} & L _ { 1 } : 2 x + 3 y + p - 3 = 0 \\ & L _ { 2 } : 2 x + 3 y + p + 3 = 0 \end{aligned}$$
where $p$ is a real number, and $C : x ^ { 2 } + y ^ { 2 } + 6 x - 10 y + 30 = 0$. STATEMENT-1 : If line $L _ { 1 }$ is a chord of circle $C$, then line $L _ { 2 }$ is not always a diameter of circle $C$.
and
STATEMENT-2 : If line $L _ { 1 }$ is a diameter of circle $C$, then line $L _ { 2 }$ is not a chord of circle $C$.
(A) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1
(B) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1
(C) STATEMENT-1 is True, STATEMENT-2 is False
(D) STATEMENT-1 is False, STATEMENT-2 is True

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]

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

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] $p = 5, h = 4, k = -3$

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 $E$ 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 $\_\_\_\_$.

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 $\_\_\_\_$ .

If the line $y = m x + 1$ meets the circle $x ^ { 2 } + y ^ { 2 } + 3 x = 0$ in two points equidistant from and on opposite sides of $x$-axis, then
(1) $3 m + 2 = 0$
(2) $3 m - 2 = 0$
(3) $2 m + 3 = 0$
(4) $2 m - 3 = 0$

If two parallel chords of a circle, having diameter 4 units, lie on the opposite sides of the center and subtend angles $\cos ^ { - 1 } \left( \frac { 1 } { 7 } \right)$ and $\sec ^ { - 1 } ( 7 )$ at the center respectively, then the distance between these chords is:
(1) $\frac { 8 } { \sqrt { 7 } }$
(2) $\frac { 16 } { 7 }$
(3) $\frac { 4 } { \sqrt { 7 } }$
(4) $\frac { 8 } { 7 }$

The sum of the squares of the lengths of the chords intercepted on the circle, $x^2 + y^2 = 16$, by the lines, $x + y = n$, $n \in N$, where $N$ is the set of all natural numbers is:
(1) 210
(2) 105
(3) 320
(4) 160

Let the latus rectum of the parabola $y ^ { 2 } = 4 x$ be the common chord to the circles $C _ { 1 }$ and $C _ { 2 }$ each of them having radius $2 \sqrt { 5 }$. Then, the distance between the centres of the circles $C _ { 1 }$ and $C _ { 2 }$ is :
(1) 12
(2) 8
(3) $8 \sqrt { 5 }$
(4) $4 \sqrt { 5 }$

If the length of the chord of the circle, $x^2 + y^2 = r^2$ $(r > 0)$ along the line, $y - 2x = 3$ is $r$, then $r^2$ is equal to:
(1) $\frac{9}{5}$
(2) 12
(3) $\frac{24}{5}$
(4) $\frac{12}{5}$

Let a circle $C : ( x - h ) ^ { 2 } + ( y - k ) ^ { 2 } = r ^ { 2 } , k > 0$, touch the $x$-axis at $( 1,0 )$. If the line $x + y = 0$ intersects the circle $C$ at $P$ and $Q$ such that the length of the chord $P Q$ is 2, then the value of $h + k + r$ is equal to $\_\_\_\_$.

The line $y = x + 1$ meets the ellipse $\frac{x^2}{4} + \frac{y^2}{2} = 1$ at two points $P$ and $Q$. If $r$ is the radius of the circle with $PQ$ as diameter then $3r^2$ is equal to
(1) 20
(2) 12
(3) 11
(4) 8

If the circle $x ^ { 2 } + y ^ { 2 } - 2 g x + 6 y - 19 c = 0 , g , c \in \mathbb { R }$ passes through the point $( 6,1 )$ and its centre lies on the line $x - 2 c y = 8$, then the length of intercept made by the circle on $x$-axis is
(1) $\sqrt { 11 }$
(2) 4
(3) 3
(4) $2 \sqrt { 23 }$

The set of all values of $a^2$ for which the line $x + y = 0$ bisects two distinct chords drawn from a point $\mathrm{P}\left(\frac{1+a}{2}, \frac{1-a}{2}\right)$ on the circle $2x^2 + 2y^2 - (1+a)x - (1-a)y = 0$, is equal to:
(1) $(8, \infty)$
(2) $(0, 4]$
(3) $(4, \infty)$
(4) $(2, 12]$

Consider two circles $C_1: x^2 + y^2 = 25$ and $C_2: (x - \alpha)^2 + y^2 = 16$, where $\alpha \in (5, 9)$. Let the angle between the two radii (one to each circle) drawn from one of the intersection points of $C_1$ and $C_2$ be $\sin^{-1}\frac{\sqrt{63}}{8}$. If the length of common chord of $C_1$ and $C_2$ is $\beta$, then the value of $(\alpha\beta)^2$ equals $\underline{\hspace{1cm}}$.

Let $C$ be a circle with radius $\sqrt { 10 }$ units and centre at the origin. Let the line $x + y = 2$ intersects the circle C at the points P and Q . Let MN be a chord of C of length 2 unit and slope - 1 . Then, a distance (in units) between the chord PQ and the chord MN is
(1) $3 - \sqrt { 2 }$
(2) $\sqrt { 2 } + 1$
(3) $\sqrt { 2 } - 1$
(4) $2 - \sqrt { 3 }$

Let the circle $C _ { 1 } : x ^ { 2 } + y ^ { 2 } - 2 ( x + y ) + 1 = 0$ and $C _ { 2 }$ be a circle having centre at $( - 1,0 )$ and radius 2 . If the line of the common chord of $\mathrm { C } _ { 1 }$ and $\mathrm { C } _ { 2 }$ intersects the $y$-axis at the point P , then the square of the distance of P from the centre of $\mathrm { C } _ { 1 }$ is :
(1) 2
(2) 1
(3) 4
(4) 6

The equation of the chord of the ellipse $\frac{x^{2}}{25} + \frac{y^{2}}{16} = 1$, whose mid-point is $(3, 1)$ is:
(1) $48x + 25y = 169$
(2) $5x + 16y = 31$
(3) $25x + 101y = 176$
(4) $4x + 122y = 134$