Two parabolas with a common vertex and with axes along $x$-axis and $y$-axis, respectively, intersect each other in the first quadrant. if the length of the latus rectum of each parabola is 3 , then the equation of the common tangent to the two parabolas is?
(1) $3 ( x + y ) + 4 = 0$
(2) $8 ( 2 x + y ) + 3 = 0$
(3) $4 ( x + y ) + 3 = 0$
(4) $x + 2 y + 3 = 0$

The equation of a tangent to the parabola, $x ^ { 2 } = 8 y$, which makes an angle $\theta$ with the positive direction of $x$-axis, is
(1) $y = x \tan \theta + 2 \cot \theta$
(2) $y = x \tan \theta - 2 \cot \theta$
(3) $x = y \cot \theta + 2 \tan \theta$
(4) $x = y \cot \theta - 2 \tan \theta$

If the eccentricity of the standard hyperbola passing through the point $( 4 , 6 )$ is 2 , then the equation of the tangent to the hyperbola at $( 4 , 6 )$ is:
(1) $2 x - 3 y + 10 = 0$
(2) $x - 2 y + 8 = 0$
(3) $3 x - 2 y = 0$
(4) $2 x - y - 2 = 0$

If the tangents on the ellipse $4x^2 + y^2 = 8$ at the points $(1,2)$ and $(a,b)$ are perpendicular to each other, then $a^2$ is equal to
(1) $\frac{2}{17}$
(2) $\frac{4}{17}$
(3) $\frac{64}{17}$
(4) $\frac{128}{17}$

The common tangent to the circles $x ^ { 2 } + y ^ { 2 } = 4$ and $x ^ { 2 } + y ^ { 2 } + 6 x + 8 y - 24 = 0$ also passes through the point:
(1) $( 4 , - 2 )$
(2) $( - 4,6 )$
(3) $( 6 , - 2 )$
(4) $( - 6,4 )$

If the tangent to the parabola $y ^ { 2 } = x$ at a point $( \alpha , \beta ) , ( \beta > 0 )$ is also a tangent to the ellipse, $x ^ { 2 } + 2 y ^ { 2 } = 1$ then $\alpha$ is equal to:
(1) $\sqrt { 2 } - 1$
(2) $2 \sqrt { 2 } + 1$
(3) $\sqrt { 2 } + 1$
(4) $2 \sqrt { 2 } - 1$

Let the tangents drawn from the origin to the circle, $x ^ { 2 } + y ^ { 2 } - 8 x - 4 y + 16 = 0$ touch it at the points $A$ and $B$. Then $( A B ) ^ { 2 }$ is equal to
(1) $\frac { 52 } { 5 }$
(2) $\frac { 56 } { 5 }$
(3) $\frac { 64 } { 5 }$
(4) $\frac { 32 } { 5 }$

If a line $y = mx + c$ is a tangent to the circle $(x - 3)^{2} + y^{2} = 1$, and it is perpendicular to a line $L_{1}$, where $L_{1}$ is the tangent to the circle $x^{2} + y^{2} = 1$ at the point $\left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)$, then
(1) $c^{2} - 7c + 6 = 0$
(2) $c^{2} + 7c + 6 = 0$
(3) $c^{2} + 6c + 7 = 0$
(4) $c^{2} - 6c + 7 = 0$

A circle touches the $y$-axis at the point $( 0,4 )$ and passes through the point $( 2,0 )$. Which of the following lines is not a tangent to this circle?
(1) $4 x - 3 y + 17 = 0$
(2) $3 x - 4 y - 24 = 0$
(3) $3 x + 4 y - 6 = 0$
(4) $4 x + 3 y - 8 = 0$

If the common tangent to the parabolas, $y ^ { 2 } = 4 x$ and $x ^ { 2 } = 4 y$ also touches the circle, $x ^ { 2 } + y ^ { 2 } = c ^ { 2 }$, then $c$ is equal to :
(1) $\frac { 1 } { 2 \sqrt { 2 } }$
(2) $\frac { 1 } { \sqrt { 2 } }$
(3) $\frac { 1 } { 4 }$
(4) $\frac { 1 } { 2 }$

A tangent is drawn to the parabola $y ^ { 2 } = 6 x$ which is perpendicular to the line $2 x + y = 1$. Which of the following points does NOT lie on it?
(1) $( 0,3 )$
(2) $( 4,5 )$
(3) $( 5,4 )$
(4) $( - 6,0 )$

A line is a common tangent to the circle $( x - 3 ) ^ { 2 } + y ^ { 2 } = 9$ and the parabola $y ^ { 2 } = 4 x$. If the two points of contact $( a , b )$ and $( c , d )$ are distinct and lie in the first quadrant, then $2 ( a + c )$ is equal to $\underline{\hspace{1cm}}$.

Let $A B C D$ be a square of side of unit length. Let a circle $C _ { 1 }$ centered at $A$ with unit radius is drawn. Another circle $C _ { 2 }$ which touches $C _ { 1 }$ and the lines $A D$ and $A B$ are tangent to it, is also drawn. Let a tangent line from the point $C$ to the circle $C _ { 2 }$ meet the side $A B$ at $E$. If the length of $E B$ is $\alpha + \sqrt { 3 } \beta$, where $\alpha , \beta$ are integers, then $\alpha + \beta$ is equal to $\_\_\_\_$.

Let $L$ be a tangent line to the parabola $y ^ { 2 } = 4 x - 20$ at (6, 2). If $L$ is also a tangent to the ellipse $\frac { x ^ { 2 } } { 2 } + \frac { y ^ { 2 } } { b } = 1$, then the value of $b$ is equal to:
(1) 11
(2) 14
(3) 16
(4) 20

The line $12 x \cos \theta + 5 y \sin \theta = 60$ is tangent to which of the following curves ?
(1) $x ^ { 2 } + y ^ { 2 } = 30$
(2) $144 x ^ { 2 } + 25 y ^ { 2 } = 3600$
(3) $x ^ { 2 } + y ^ { 2 } = 169$
(4) $25 x ^ { 2 } + 12 y ^ { 2 } = 3600$

If $y = m _ { 1 } x + c _ { 1 }$ and $y = m _ { 2 } x + c _ { 2 } , \quad m _ { 1 } \neq m _ { 2 }$ are two common tangents of circle $x ^ { 2 } + y ^ { 2 } = 2$ and parabola $y ^ { 2 } = x$, then the value of $8 \quad m _ { 1 } \quad m _ { 2 }$ is equal to
(1) $3 \sqrt { 2 } - 4$
(2) $6 \sqrt { 2 } - 4$
(3) $- 5 + 6 \sqrt { 2 }$
(4) $3 + 4 \sqrt { 2 }$

The equation of a common tangent to the parabolas $y = x ^ { 2 }$ and $y = -(x - 2) ^ { 2 }$ is
(1) $y = 4 x - 2$
(2) $y = 4 x - 1$
(3) $y = 4 x + 1$
(4) $y = 4 x + 2$

Let the normal at the point $P$ on the parabola $y ^ { 2 } = 6 x$ pass through the point $( 5 , - 8 )$. If the tangent at $P$ to the parabola intersects its directrix at the point $Q$, then the ordinate of the point $Q$ is
(1) $\frac { - 9 } { 4 }$
(2) $\frac { 9 } { 4 }$
(3) $\frac { - 5 } { 2 }$
(4) $- 3$

Let $P ( a , b )$ be a point on the parabola $y ^ { 2 } = 8 x$ such that the tangent at $P$ passes through the centre of the circle $x ^ { 2 } + y ^ { 2 } - 10 x - 14 y + 65 = 0$. Let $A$ be the product of all possible values of $a$ and $B$ be the product of all possible values of $b$. Then the value of $A + B$ is equal to
(1) 0
(2) 25
(3) 40
(4) 65

A circle $C _ { 1 }$ passes through the origin $O$ and has diameter 4 on the positive $x$-axis. The line $y = 2 x$ gives a chord $O A$ of a circle $C _ { 1 }$. Let $C _ { 2 }$ be the circle with $O A$ as a diameter. If the tangent to $C _ { 2 }$ at the point $A$ meets the $x$-axis at $P$ and $y$-axis at $Q$, then $Q A : A P$ is equal to
(1) $1 : 4$
(2) $1 : 5$
(3) $2 : 5$
(4) $1 : 3$

Two tangent lines $l _ { 1 }$ and $l _ { 2 }$ are drawn from the point $( 2,0 )$ to the parabola $2 y ^ { 2 } = - x$. If the lines $l _ { 1 }$ and $l _ { 2 }$ are also tangent to the circle $( x - 5 ) ^ { 2 } + y ^ { 2 } = r$, then $17 r ^ { 2 }$ is equal to $\_\_\_\_$.

Let the tangents at the points $P$ and $Q$ on the ellipse $\frac { x ^ { 2 } } { 2 } + \frac { y ^ { 2 } } { 4 } = 1$ meet at the point $R ( \sqrt { 2 } , 2 \sqrt { 2 } - 2 )$. If $S$ is the focus of the ellipse on its negative major axis, then $S P ^ { 2 } + S Q ^ { 2 }$ is equal to $\_\_\_\_$.

The number of common tangents, to the circles $x ^ { 2 } + y ^ { 2 } - 18 x - 15 y + 131 = 0$ and $x ^ { 2 } + y ^ { 2 } - 6 x - 6 y - 7 = 0$, is
(1) 3
(2) 1
(3) 4
(4) 2

Points $P ( - 3,2 ) , Q ( 9,10 )$ and $R ( \alpha , 4 )$ lie on a circle $C$ with $P R$ as its diameter. The tangents to $C$ at the points $Q$ and $R$ intersect at the point $S$. If $S$ lies on the line $2 x - k y = 1$, then $k$ is equal to $\_\_\_\_$.

Let the tangents at the points $A ( 4 , - 11 )$ and $B ( 8 , - 5 )$ on the circle $x ^ { 2 } + y ^ { 2 } - 3 x + 10 y - 15 = 0$, intersect at the point $C$. Then the radius of the circle, whose centre is $C$ and the line joining $A$ and $B$ is its tangent, is equal to
(1) $\frac { 3 \sqrt { 3 } } { 4 }$
(2) $2 \sqrt { 13 }$
(3) $\sqrt { 13 }$
(4) $\frac { 2 \sqrt { 13 } } { 3 }$