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) $4x + 122y = 134$

If the midpoint of a chord of the ellipse $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 4 } = 1$ is $( \sqrt { 2 } , 4 / 3 )$, and the length of the chord is $\frac { 2 \sqrt { \alpha } } { 3 }$, then $\alpha$ is :
(1) 20
(2) 22
(3) 18
(4) 26

Let a circle $C$ pass through the points $( 4,2 )$ and $( 0,2 )$, and its centre lie on $3 x + 2 y + 2 = 0$. Then the length of the chord, of the circle $C$, whose mid-point is $( 1,2 )$, is :
(1) $\sqrt { 3 }$
(2) $2 \sqrt { 2 }$
(3) $2 \sqrt { 3 }$
(4) $4 \sqrt { 2 }$

On the coordinate plane, a circle with radius 12 intersects the line $x + y = 0$ at two points, and the distance between these two points is 8. If this circle intersects the line $x + y = 24$ at points $P$ and $Q$, then the length of segment $\overline { P Q }$ is $\_\_\_\_$ (14)$\sqrt { (15) }$. (Express as a simplified radical)

As shown in the figure, $L$ is a line passing through the origin $O$ on the coordinate plane, $\Gamma$ is a circle centered at $O$, and $L$ and $\Gamma$ have one intersection point $A ( 3,4 )$. It is known that $B , C$ are two distinct points on $\Gamma$ satisfying $\overrightarrow { B C } = \overrightarrow { O A }$. Select the correct options.
(1) The other intersection point of $L$ and $\Gamma$ is $( - 4 , - 3 )$
(2) The slope of line $B C$ is $\frac { 3 } { 4 }$
(3) $\angle A O C = 60 ^ { \circ }$
(4) The area of $\triangle A B C$ is $\frac { 25 \sqrt { 3 } } { 2 }$
(5) $B$ and $C$ are in the same quadrant

On the coordinate plane, let $\Gamma$ be a circle with center at the origin, and $P$ be one of the intersection points of $\Gamma$ and the $x$-axis. It is known that the line passing through $P$ with slope $\frac{1}{2}$ intersects $\Gamma$ at another point $Q$, and $\overline{PQ} = 1$. Then the radius of $\Gamma$ is . (Express as a simplified radical)