A hyperbola has its centre at the origin, passes through the point $(4,2)$ and has transverse axis of length 4 along the $x$-axis. Then the eccentricity of the hyperbola is:
(1) $\sqrt{3}$
(2) $\frac{3}{2}$
(3) $\frac{2}{\sqrt{3}}$
(4) 2

If a hyperbola passes through the point $P(10, 16)$, and it has vertices at $(\pm 6, 0)$, then the equation of the normal to it at $P$ is.
(1) $3x + 4y = 94$
(2) $2x + 5y = 100$
(3) $x + 2y = 42$
(4) $x + 3y = 58$

A line parallel to the straight line $2x - y = 0$ is tangent to the hyperbola $\frac{x^{2}}{4} - \frac{y^{2}}{2} = 1$ at the point $(x_{1}, y_{1})$. Then $x_{1}^{2} + 5y_{1}^{2}$ is equal to
(1) 6
(2) 8
(3) 10
(4) 5

If the distance between the foci of an ellipse is 6 and the distance between its directrix is 12, then the length of its latus rectum is
(1) $\sqrt { 3 }$
(2) $3 \sqrt { 2 }$
(3) $\frac { 3 } { \sqrt { 2 } }$
(4) $2 \sqrt { 3 }$

If $e _ { 1 }$ and $e _ { 2 }$ are the eccentricities of the ellipse $\frac { x ^ { 2 } } { 18 } + \frac { y ^ { 2 } } { 4 } = 1$ and the hyperbola $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 4 } = 1$ respectively and $\left( e _ { 1 } , e _ { 2 } \right)$ is a point on the ellipse $15 x ^ { 2 } + 3 y ^ { 2 } = k$, then the value of $k$ is equal to
(1) 16
(2) 17
(3) 15
(4) 14

A hyperbola having the transverse axis of length, $\sqrt { 2 }$ has the same foci as that of the ellipse, $3 x ^ { 2 } + 4 y ^ { 2 } = 12$ then this hyperbola does not pass through which of the following points?
(1) $\left( \frac { 1 } { \sqrt { 2 } } , 0 \right)$
(2) $\left( - \sqrt { \frac { 3 } { 2 } } , 1 \right)$
(3) $\left( 1 , - \frac { 1 } { \sqrt { 2 } } \right)$
(4) $\left( \sqrt { \frac { 3 } { 2 } } , \frac { 1 } { \sqrt { 2 } } \right)$

Let $e _ { 1 }$ and $e _ { 2 }$ be the eccentricities of the ellipse $\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( b < 5 )$ and the hyperbola $\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ respectively satisfying $\mathrm { e } _ { 1 } \mathrm { e } _ { 2 } = 1$. If $\alpha$ and $\beta$ are the distances between the foci of the ellipse and the foci of the hyperbola respectively, then the ordered pair $( \alpha , \beta )$ is equal to:
(1) $( 8,10 )$
(2) $\left( \frac { 20 } { 3 } , 12 \right)$
(3) $( 8,12 )$
(4) $\left( \frac { 24 } { 5 } , 10 \right)$

Let $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > b )$ be a given ellipse, length of whose latus rectum is 10 . If its eccentricity is the maximum value of the function, $\phi ( t ) = \frac { 5 } { 12 } + t - t ^ { 2 }$, then $a ^ { 2 } + b ^ { 2 }$ is equal to:
(1) 145
(2) 116
(3) 126
(4) 135

If the normal at an end of latus rectum of an ellipse passes through an extremity of the minor axis, then the eccentricity e of the ellipse satisfies:
(1) $\mathrm{e}^{4}+2\mathrm{e}^{2}-1=0$
(2) $\mathrm{e}^{2}+\mathrm{e}-1=0$
(3) $\mathrm{e}^{4}+\mathrm{e}^{2}-1=0$
(4) $\mathrm{e}^{2}+2\mathrm{e}-1=0$

The length of the minor axis (along $y$-axis) of an ellipse in the standard form is $\frac { 4 } { \sqrt { 3 } }$. If this ellipse touches the line $x + 6 y = 8$ then its eccentricity is:
(1) $\frac { 1 } { 2 } \sqrt { \frac { 11 } { 3 } }$
(2) $\sqrt { \frac { 5 } { 6 } }$
(3) $\frac { 1 } { 2 } \sqrt { \frac { 5 } { 3 } }$
(4) $\frac { 1 } { 3 } \sqrt { \frac { 11 } { 3 } }$

Let $P ( 3,3 )$ be a point on the hyperbola, $\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1$. If the normal to it at $P$ intersects the $x$-axis at $( 9,0 )$ and $e$ is its eccentricity, then the ordered pair $\left( a ^ { 2 } , e ^ { 2 } \right)$ is equal to:
(1) $\left( \frac { 9 } { 2 } , 3 \right)$
(2) $\left( \frac { 3 } { 2 } , 2 \right)$
(3) $\left( \frac { 9 } { 2 } , 2 \right)$
(4) $( 9,3 )$

If the line $y = mx + c$ is a common tangent to the hyperbola $\frac{x^2}{100} - \frac{y^2}{64} = 1$ and the circle $x^2 + y^2 = 36$, then which one of the following is true?
(1) $c^2 = 369$
(2) $5m = 4$
(3) $4c^2 = 369$
(4) $8m + 5 = 0$

If $3 x + 4 y = 12 \sqrt { 2 }$ is a tangent to the ellipse $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { 9 } = 1$ for some $a \in R$, then the distance between the foci of the ellipse is
(1) $2 \sqrt { 7 }$
(2) 4
(3) $2 \sqrt { 5 }$
(4) $2 \sqrt { 2 }$

For some $\theta \in \left( 0 , \frac { \pi } { 2 } \right)$, if the eccentricity of the hyperbola, $x ^ { 2 } - y ^ { 2 } \sec ^ { 2 } \theta = 10$ is $\sqrt { 5 }$ times the eccentricity of the ellipse, $x ^ { 2 } \sec ^ { 2 } \theta + y ^ { 2 } = 5$, then the length of the latus rectum of the ellipse, is
(1) $2 \sqrt { 6 }$
(2) $\sqrt { 30 }$
(3) $\frac { 2 \sqrt { 5 } } { 3 }$
(4) $\frac { 4 \sqrt { 5 } } { 3 }$

If two tangents drawn from a point $P$ to the parabola $y ^ { 2 } = 16 ( x - 3 )$ are at right angles, then the locus of point $P$ is: (1) $x + 4 = 0$ (2) $x + 2 = 0$ (3) $x + 3 = 0$ (4) $x + 1 = 0$

Let $E _ { 1 } : \frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 , a > b$. Let $E _ { 2 }$ be another ellipse such that it touches the end points of major axis of $E _ { 1 }$ and the foci of $E _ { 2 }$ are the end points of minor axis of $E _ { 1 }$. If $E _ { 1 }$ and $E _ { 2 }$ have same eccentricities, then its value is:
(1) $\frac { - 1 + \sqrt { 5 } } { 2 }$
(2) $\frac { - 1 + \sqrt { 8 } } { 2 }$
(3) $\frac { - 1 + \sqrt { 3 } } { 2 }$
(4) $\frac { - 1 + \sqrt { 6 } } { 2 }$

If the points of intersection of the ellipse $\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ and the circle $x ^ { 2 } + y ^ { 2 } = 4b , b > 4$ lie on the curve $y ^ { 2 } = 3x ^ { 2 }$, then $b$ is equal to :
(1) 12
(2) 5
(3) 6
(4) 10

Let a line $L : 2 x + y = k , k > 0$ be a tangent to the hyperbola $x ^ { 2 } - y ^ { 2 } = 3$. If $L$ is also a tangent to the parabola $y ^ { 2 } = \alpha x$, then $\alpha$ is equal to:
(1) 12
(2) - 12
(3) 24
(4) - 24

A hyperbola passes through the foci of the ellipse $\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 16 } = 1$ and its transverse and conjugate axes coincide with major and minor axes of the ellipse, respectively. If the product of their eccentricities is one, then the equation of the hyperbola is:
(1) $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 16 } = 1$
(2) $x ^ { 2 } - y ^ { 2 } = 9$
(3) $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 25 } = 1$
(4) $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 4 } = 1$

Consider a hyperbola $H : x ^ { 2 } - 2 y ^ { 2 } = 4$. Let the tangent at a point $P ( 4 , \sqrt { 6 } )$ meet the $x$-axis at $Q$ and latus rectum at $R \left( x _ { 1 } , y _ { 1 } \right) , x _ { 1 } > 0$. If $F$ is a focus of $H$ which is nearer to the point $P$, then the area of $\triangle QFR$ (in sq. units) is equal to
(1) $4 \sqrt { 6 }$
(2) $\sqrt { 6 } - 1$
(3) $\frac { 7 } { \sqrt { 6 } } - 2$
(4) $4 \sqrt { 6 } - 1$

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

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) $( 3 \sqrt { 6 } , 6 \sqrt { 2 } )$