The equation of a tangent to the hyperbola, $4 x ^ { 2 } - 5 y ^ { 2 } = 20$, parallel to the line $x - y = 2$, is
(1) $x - y + 7 = 0$
(2) $x - y - 3 = 0$
(3) $x - y + 1 = 0$
(4) $x - y + 9 = 0$

If the line $y = m x + 7 \sqrt { 3 }$ is normal to the hyperbola $\frac { x ^ { 2 } } { 24 } - \frac { y ^ { 2 } } { 18 } = 1$, then a value of $m$ is:
(1) $\frac { \sqrt { 5 } } { 2 }$
(2) $\frac { 3 } { \sqrt { 5 } }$
(3) $\frac { \sqrt { 15 } } { 2 }$
(4) $\frac { 2 } { \sqrt { 5 } }$

If $y = m x + 4$ is a tangent to both the parabolas, $y ^ { 2 } = 4 x$ and $x ^ { 2 } = 2 b y$, then $b$ is equal to
(1) $-32$
(2) $-64$
(3) $-128$
(4) 128

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

Let $x = 4$ be a directrix to an ellipse whose centre is at the origin and its eccentricity is $\frac { 1 } { 2 }$. If $P ( 1 , \beta ) , \beta > 0$ is a point on this ellipse, then the equation of the normal to it at $P$ is
(1) $4 x - 3 y = 2$
(2) $8 x - 2 y = 5$
(3) $7 x - 4 y = 1$
(4) $4 x - 2 y = 1$

Let $L _ { 1 }$ be a tangent to the parabola $y ^ { 2 } = 4 ( x + 1 )$ and $L _ { 2 }$ be a tangent to the parabola $y ^ { 2 } = 8 ( x + 2 )$ such that $L _ { 1 }$ and $L _ { 2 }$ intersect at right angles. Then $L _ { 1 }$ and $L _ { 2 }$ meet on the straight line:
(1) $x + 3 = 0$
(2) $2 x + 1 = 0$
(3) $x + 2 = 0$
(4) $x + 2 y = 0$

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

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 $\lambda x - 2 y = \mu$ 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 $m$ 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 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

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 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 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) $( 15,13 )$

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 $\_\_\_\_$ .

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 $\_\_\_\_$ .