Let the eccentricity of the hyperbola $H : \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ be $\sqrt{\frac{5}{2}}$ and length of its latus rectum be $6\sqrt{2}$. If $y = 2x + c$ is a tangent to the hyperbola $H$, then the value of $c^2$ is equal to
(1) 18
(2) 20
(3) 24
(4) 32

Let the focal chord of the parabola $P : y ^ { 2 } = 4 x$ along the line $L : y = m x + c , m > 0$ meet the parabola at the points $M$ and $N$. Let the line $L$ be a tangent to the hyperbola $H : x ^ { 2 } - y ^ { 2 } = 4$. If $O$ is the vertex of $P$ and $F$ is the focus of $H$ on the positive $x$-axis, then the area of the quadrilateral $O M F N$ is
(1) $2 \sqrt { 6 }$
(2) $2 \sqrt { 14 }$
(3) $4 \sqrt { 6 }$
(4) $4 \sqrt { 14 }$

Let the maximum area of the triangle that can be inscribed in the ellipse $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { 4 } = 1 , a > 2$, having one of its vertices at one end of the major axis of the ellipse and one of its sides parallel to the $y$-axis, be $6 \sqrt { 3 }$. Then the eccentricity of the ellipse is:
(1) $\frac { \sqrt { 3 } } { 2 }$
(2) $\frac { 1 } { 2 }$
(3) $\frac { 1 } { \sqrt { 2 } }$
(4) $\frac { \sqrt { 3 } } { 4 }$

Let $P Q$ be a focal chord of the parabola $y ^ { 2 } = 4 x$ such that it subtends an angle of $\frac { \pi } { 2 }$ at the point $( 3,0 )$. Let the line segment $P Q$ be also a focal chord of the ellipse $E : \frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 , a ^ { 2 } > b ^ { 2 }$. If $e$ is the eccentricity of the ellipse $E$, then the value of $\frac { 1 } { e ^ { 2 } }$ is equal to
(1) $1 + \sqrt { 2 }$
(2) $3 + 2 \sqrt { 2 }$
(3) $1 + 2 \sqrt { 3 }$
(4) $4 + 5 \sqrt { 3 }$

The acute angle between the pair of tangents drawn to the ellipse $2 x ^ { 2 } + 3 y ^ { 2 } = 5$ from the point $(1, 3)$ is
(1) $\tan ^ { - 1 } \frac { 16 } { 7 \sqrt { 5 } }$
(2) $\tan ^ { - 1 } \frac { 24 } { 7 \sqrt { 5 } }$
(3) $\tan ^ { - 1 } \frac { 32 } { 7 \sqrt { 5 } }$
(4) $\tan ^ { - 1 } \frac { 3 + 8 \sqrt { 5 } } { 35 }$

Let $\lambda x - 2 y = \mu$ be a tangent to the hyperbola $a ^ { 2 } x ^ { 2 } - y ^ { 2 } = b ^ { 2 }$. Then $\left( \frac { \lambda } { a } \right) ^ { 2 } - \left( \frac { \mu } { b } \right) ^ { 2 }$ is equal to
(1) - 2
(2) - 4
(3) 2
(4) 4

If the line $x - 1 = 0$, is a directrix of the hyperbola $k x ^ { 2 } - y ^ { 2 } = 6$, then the hyperbola passes through the point
(1) $\left( - 2 \sqrt { 5 } , 6 \right)$
(2) $\left( - \sqrt { 5 } , 3 \right)$
(3) $\left( \sqrt { 5 } , - 2 \right)$
(4) $\left( 2 \sqrt { 5 } , 3 \sqrt { 6 } \right)$

If the equation of the parabola, whose vertex is at $( 5,4 )$ and the directrix is $3 x + y - 29 = 0$, is $x ^ { 2 } + a y ^ { 2 } + b x y + c x + d y + k = 0$, then $a + b + c + d + k$ is equal to
(1) 575
(2) - 575
(3) 576
(4) - 576

Let $P _ { 1 }$ be a parabola with vertex $( 3,2 )$ and focus $( 4,4 )$ and $P _ { 2 }$ be its mirror image with respect to the line $x + 2 y = 6$. Then the directrix of $P _ { 2 }$ is $x + 2 y =$ $\_\_\_\_$.

Let the hyperbola $H : \frac { x ^ { 2 } } { a ^ { 2 } } - y ^ { 2 } = 1$ and the ellipse $E : 3 x ^ { 2 } + 4 y ^ { 2 } = 12$ be such that the length of latus rectum of $H$ is equal to the length of latus rectum of $E$. If $e _ { H }$ and $e _ { E }$ are the eccentricities of $H$ and $E$ respectively, then the value of $12 \left( e _ { H } ^ { 2 } + e _ { E } ^ { 2 } \right)$ is equal to $\_\_\_\_$.

Let the equation of two diameters of a circle $x ^ { 2 } + y ^ { 2 } - 2x + 2fy + 1 = 0$ be $2px - y = 1$ and $2x + py = 4p$. Then the slope $m \in (0,\infty)$ of the tangent to the hyperbola $3x ^ { 2 } - y ^ { 2 } = 3$ passing through the centre of the circle is equal to $\_\_\_\_$.

Let H be the hyperbola, whose foci are $(1 \pm \sqrt{2}, 0)$ and eccentricity is $\sqrt{2}$. Then the length of its latus rectum is:
(1) 3
(2) $\frac{5}{2}$
(3) 2
(4) $\frac{3}{2}$

The vertices of a hyperbola H are $( \pm 6,0 )$ and its eccentricity is $\frac { \sqrt { 5 } } { 2 }$. Let N be the normal to H at a point in the first quadrant and parallel to the line $\sqrt { 2 } x + y = 2 \sqrt { 2 }$. If $d$ is the length of the line segment of N between H and the $y$-axis then $d ^ { 2 }$ is equal to $\_\_\_\_$ .

Let the eccentricity of an ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ is reciprocal to that of the hyperbola $2x^{2} - 2y^{2} = 1$. If the ellipse intersects the hyperbola at right angles, then square of length of the latus-rectum of the ellipse is $\_\_\_\_$.

Let $P \left( \frac { 2 \sqrt { 3 } } { \sqrt { 7 } } , \frac { 6 } { \sqrt { 7 } } \right) , Q , R$ and $S$ be four points on the ellipse $9 x ^ { 2 } + 4 y ^ { 2 } = 36$. Let $P Q$ and $R S$ be mutually perpendicular and pass through the origin. If $\frac { 1 } { ( P Q ) ^ { 2 } } + \frac { 1 } { ( R S ) ^ { 2 } } = \frac { p } { q }$, where $p$ and $q$ are coprime, then $p + q$ is equal to
(1) 147
(2) 143
(3) 137
(4) 157

The equations of two sides of a variable triangle are $x = 0$ and $y = 3$, and its third side is a tangent to the parabola $y ^ { 2 } = 6 x$. The locus of its circumcentre is:
(1) $4 y ^ { 2 } - 18 y - 3 x - 18 = 0$
(2) $4 y ^ { 2 } + 18 y + 3 x + 18 = 0$
(3) $4 y ^ { 2 } - 18 y + 3 x + 18 = 0$
(4) $4 y ^ { 2 } - 18 y - 3 x + 18 = 0$

If the foci of a hyperbola are same as that of the ellipse $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 25 } = 1$ and the eccentricity of the hyperbola is $\frac { 15 } { 8 }$ times the eccentricity of the ellipse, then the smaller focal distance of the point $\left(\sqrt { 2 } , \frac { 14 } { 3 } \sqrt { \frac { 2 } { 5 } }\right)$ on the hyperbola is equal to
(1) $7 \sqrt { \frac { 2 } { 5 } } - \frac { 8 } { 3 }$
(2) $14 \sqrt { \frac { 2 } { 5 } } - \frac { 4 } { 3 }$
(3) $14 \sqrt { \frac { 2 } { 5 } } - \frac { 16 } { 3 }$
(4) $7 \sqrt { \frac { 2 } { 5 } } + \frac { 8 } { 3 }$

Let the foci of a hyperbola $H$ coincide with the foci of the ellipse $E : \frac { ( x - 1 ) ^ { 2 } } { 100 } + \frac { ( y - 1 ) ^ { 2 } } { 75 } = 1$ and the eccentricity of the hyperbola $H$ be the reciprocal of the eccentricity of the ellipse $E$. If the length of the transverse axis of $H$ is $\alpha$ and the length of its conjugate axis is $\beta$, then $3 \alpha ^ { 2 } + 2 \beta ^ { 2 }$ is equal to
(1) 237
(2) 242
(3) 205
(4) 225

If the length of the minor axis of ellipse is equal to half of the distance between the foci, then the eccentricity of the ellipse is :
(1) $\frac { \sqrt { 5 } } { 3 }$
(2) $\frac { \sqrt { 3 } } { 2 }$
(3) $\frac { 1 } { \sqrt { 3 } }$
(4) $\frac { 2 } { \sqrt { 5 } }$

Let $e _ { 1 }$ be the eccentricity of the hyperbola $\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 9 } = 1$ and $e _ { 2 }$ be the eccentricity of the ellipse $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 , a > b$, which passes through the foci of the hyperbola. If $e _ { 1 } e _ { 2 } = 1$, then the length of the chord of the ellipse parallel to the x -axis and passing through $( 0,2 )$ is:
(1) $4 \sqrt { 5 }$
(2) $\frac { 8 \sqrt { 5 } } { 3 }$
(3) $\frac { 10 \sqrt { 5 } } { 3 }$
(4) $3 \sqrt { 5 }$

Let $P$ be a point on the hyperbola $H: \frac{x^2}{9} - \frac{y^2}{4} = 1$, in the first quadrant such that the area of triangle formed by $P$ and the two foci of $H$ is $2\sqrt{13}$. Then, the square of the distance of $P$ from the origin is
(1) 18
(2) 26
(3) 22
(4) 20

Consider a hyperbola H having centre at the origin and foci on the x-axis. Let $\mathrm { C } _ { 1 }$ be the circle touching the hyperbola H and having the centre at the origin. Let $\mathrm { C } _ { 2 }$ be the circle touching the hyperbola H at its vertex and having the centre at one of its foci. If areas (in sq units) of $C _ { 1 }$ and $C _ { 2 }$ are $36 \pi$ and $4 \pi$, respectively, then the length (in units) of latus rectum of H is
(1) $\frac { 14 } { 3 }$
(2) $\frac { 28 } { 3 }$
(3) $\frac { 11 } { 3 }$
(4) $\frac { 10 } { 3 }$

Let $H : \frac { - x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ be the hyperbola, whose eccentricity is $\sqrt { 3 }$ and the length of the latus rectum is $4 \sqrt { 3 }$. Suppose the point $( \alpha , 6 ) , \alpha > 0$ lies on $H$. If $\beta$ is the product of the focal distances of the point $( \alpha , 6 )$, then $\alpha ^ { 2 } + \beta$ is equal to
(1) 172
(2) 171
(3) 169
(4) 170

Let $P$ be a parabola with vertex $(2, 3)$ and directrix $2x + y = 6$. Let an ellipse $E : \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$, $a > b$ of eccentricity $\dfrac{1}{\sqrt{2}}$ pass through the focus of the parabola $P$. Then the square of the length of the latus rectum of $E$, is
(1) $\dfrac{385}{8}$
(2) $\dfrac{347}{8}$
(3) $\dfrac{512}{25}$
(4) $\dfrac{656}{25}$

The length of the latus rectum and directrices of a hyperbola with eccentricity e are 9 and $x = \pm \frac { 4 } { \sqrt { 13 } }$, respectively. Let the line $y - \sqrt { 3 } x + \sqrt { 3 } = 0$ touch this hyperbola at $( x _ { 0 } , y _ { 0 } )$. If m is the product of the focal distances of the point $\left( x _ { 0 } , y _ { 0 } \right)$, then $4 \mathrm { e } ^ { 2 } + \mathrm { m }$ is equal to $\_\_\_\_$