The locus of the midpoints of the chord of the circle, $x ^ { 2 } + y ^ { 2 } = 25$ which is tangent to the hyperbola, $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 16 } = 1$ is:

The locus of the point of intersection of the lines $( \sqrt { 3 } ) k x + k y - 4 \sqrt { 3 } = 0$ and $\sqrt { 3 } x - y - 4 ( \sqrt { 3 } ) k = 0$ is a conic, whose eccentricity is

Let $P ( a \sec \theta , b \tan \theta )$ and $Q ( a \sec \phi , b \tan \phi )$ where $\theta + \phi = \frac { \pi } { 2 }$, be two points on the hyperbola $\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1$. If the ordinate of the point of intersection of normals at $P$ and $Q$ is $- k \left( \frac { a ^ { 2 } + b ^ { 2 } } { 2 b } \right)$, then $k$ is equal to

If $m$ is the slope of a common tangent to the curves $\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 9 } = 1$ and $x ^ { 2 } + y ^ { 2 } = 12$, then $12 \mathrm {~m} ^ { 2 }$ is equal to
(1) 6
(2) 9
(3) 10
(4) 12

The locus of the mid-point of the line segment joining the point $( 4,3 )$ and the points on the ellipse $x ^ { 2 } + 2 y ^ { 2 } = 4$ is an ellipse with eccentricity
(1) $\frac { \sqrt { 3 } } { 2 }$
(2) $\frac { 1 } { 2 \sqrt { 2 } }$
(3) $\frac { 1 } { \sqrt { 2 } }$
(4) $\frac { 1 } { 2 }$

Let a line $L$ pass through the point of intersection of the lines $b x + 10 y - 8 = 0$ and $2 x - 3 y = 0$, $b \in R - \left\{ \frac { 4 } { 3 } \right\}$. If the line $L$ also passes through the point $( 1,1 )$ and touches the circle $17 \left( x ^ { 2 } + y ^ { 2 } \right) = 16$, then the eccentricity of the ellipse $\frac { x ^ { 2 } } { 5 } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ is
(1) $\frac { 2 } { \sqrt { 5 } }$
(2) $\sqrt { \frac { 3 } { 5 } }$
(3) $\frac { 1 } { \sqrt { 5 } }$
(4) $\sqrt { \frac { 2 } { 5 } }$

Let the hyperbola $H : \frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ pass through the point $( 2 \sqrt { 2 } , - 2 \sqrt { 2 } )$. A parabola is drawn whose focus is same as the focus of $H$ with positive abscissa and the directrix of the parabola passes through the other focus of $H$. If the length of the latus rectum of the parabola is $e$ times the length of the latus rectum of $H$, where $e$ is the eccentricity of $H$, then which of the following points lies on the parabola?
(1) $( 2 \sqrt { 3 } , 3 \sqrt { 2 } )$
(2) $( 3 \sqrt { 3 } , - 6 \sqrt { 2 } )$
(3) $( \sqrt { 3 } , - \sqrt { 6 } )$
(4) $( 3 \sqrt { 6 } , 6 \sqrt { 2 } )$

The normal to the hyperbola $\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { 9 } = 1$ at the point $( 8,3 \sqrt { 3 } )$ on it passes through the point
(1) $( 15 , - 2 \sqrt { 3 } )$
(2) $( 9,2 \sqrt { 3 } )$
(3) $( - 1,9 \sqrt { 3 } )$
(4) $( - 1,6 \sqrt { 3 } )$

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 tangent drawn to the parabola $y ^ { 2 } = 24 x$ at the point $( \alpha , \beta )$ is perpendicular to the line $2 x + 2 y = 5$. Then the normal to the hyperbola $\frac { x ^ { 2 } } { \alpha ^ { 2 } } - \frac { y ^ { 2 } } { \beta ^ { 2 } } = 1$ at the point $( \alpha + 4 , \beta + 4 )$ does NOT pass through the point:
(1) $( 25,10 )$
(2) $( 20,12 )$
(3) $( 30,8 )$
(4) $( 15,13 )$

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 ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ meets the line $\frac{x}{7} + \frac{y}{2\sqrt{6}} = 1$ on the $x$-axis and the line $\frac{x}{7} - \frac{y}{2\sqrt{6}} = 1$ on the $y$-axis, then the eccentricity of the ellipse is
(1) $\frac{5}{7}$
(2) $\frac{2\sqrt{6}}{7}$
(3) $\frac{3}{7}$
(4) $\frac{2\sqrt{5}}{7}$

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 tangents drawn at the points $P$ and $Q$ on the parabola $y^2 = 2x - 3$ intersect at the point $R(0, 1)$, then the orthocentre of the triangle $PQR$ is
(1) $(0, 1)$
(2) $(2, -1)$
(3) $(6, 3)$
(4) $(2, 1)$

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 the foci of the ellipse $\frac{x^2}{16} + \frac{y^2}{7} = 1$ and the hyperbola $\frac{x^2}{144} - \frac{y^2}{\alpha} = \frac{1}{25}$ coincide. Then the length of the latus rectum of the hyperbola is:
(1) $\frac{32}{9}$
(2) $\frac{18}{5}$
(3) $\frac{27}{4}$
(4) $\frac{27}{10}$

A common tangent $T$ to the curves $C _ { 1 } : \frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 9 } = 1$ and $C _ { 2 } : \frac { x ^ { 2 } } { 42 } - \frac { y ^ { 2 } } { 143 } = 1$ does not pass through the fourth quadrant. If $T$ touches $C _ { 1 }$ at $\left( x _ { 1 } , y _ { 1 } \right)$ and $C _ { 2 }$ at $\left( x _ { 2 } , y _ { 2 } \right)$, then $\left| 2 x _ { 1 } + x _ { 2 } \right|$ is equal to $\_\_\_\_$ .

An ellipse $E : \frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ passes through the vertices of the hyperbola $H : \frac { x ^ { 2 } } { 49 } - \frac { y ^ { 2 } } { 64 } = - 1$. Let the major and minor axes of the ellipse $E$ coincide with the transverse and conjugate axes of the hyperbola $H$. Let the product of the eccentricities of $E$ and $H$ be $\frac { 1 } { 2 }$. If $l$ is the length of the latus rectum of the ellipse $E$, then the value of $113 l$ is equal to $\_\_\_\_$ .

Let the common tangents to the curves $4 \left( x ^ { 2 } + y ^ { 2 } \right) = 9$ and $y ^ { 2 } = 4 x$ intersect at the point $Q$. Let an ellipse, centered at the origin $O$, has lengths of semi-minor and semi-major axes equal to $OQ$ and 6, respectively. If $e$ and $l$ respectively denote the eccentricity and the length of the latus rectum of this ellipse, then $\frac { l } { e ^ { 2 } }$ is equal to $\_\_\_\_$.

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 a tangent to the curve $y ^ { 2 } = 24 x$ meet the curve $x y = 2$ at the points $A$ and $B$. Then the midpoints of such line segments $A B$ lie on a parabola with the
(1) directrix $4 x = 3$
(2) directrix $4 x = - 3$
(3) Length of latus rectum $\frac { 3 } { 2 }$
(4) Length of latus rectum 2