The equation of a tangent to the parabola $y ^ { 2 } = 8 x$ is $y = x + 2$. The point on this line from which the other tangent to the parabola is perpendicular to the given tangent is
(1) $( - 1,1 )$
(2) $( 0,2 )$
(3) $( 2,4 )$
(4) $( - 2,0 )$

The number of common tangents of the circles given by $x ^ { 2 } + y ^ { 2 } - 8 x - 2 y + 1 = 0$ and $x ^ { 2 } + y ^ { 2 } + 6 x + 8 y = 0$ is
(1) one
(2) four
(3) two
(4) three

Given: A circle, $2x^2 + 2y^2 = 5$ and a parabola, $y^2 = 4\sqrt{5}x$. Statement-I: An equation of a common tangent to these curves is $y = x + \sqrt{5}$. Statement-II: If the line, $y = mx + \frac{\sqrt{5}}{m}$ $(m \neq 0)$ is their common tangent, then $m$ satisfies $m^4 - 3m^2 + 2 = 0$.
(1) Statement-I is true; Statement-II is false.
(2) Statement-I is false; Statement-II is true.
(3) Statement-I is true; Statement-II is true; Statement-II is a correct explanation for Statement-I.
(4) Statement-I is true; Statement-II is true; Statement-II is not a correct explanation for Statement-I.

The number of common tangents to the circles $x^2 + y^2 - 4x - 6y - 12 = 0$ and $x^2 + y^2 + 6x + 18y + 26 = 0$, is:
(1) 1
(2) 2
(3) 3
(4) 4

The number of common tangents to the circles $x ^ { 2 } + y ^ { 2 } - 4 x - 6 y - 12 = 0$ and $x ^ { 2 } + y ^ { 2 } + 6 x + 18 y + 26 = 0$, is
(1) 4
(2) 1
(3) 2
(4) 3

Equation of the tangent to the circle, at the point $( 1 , - 1 )$, whose center, is the point of intersection of the straight lines $x - y = 1$ and $2 x + y = 3$ is:
(1) $x + 4 y + 3 = 0$
(2) $3 x - y - 4 = 0$
(3) $x - 3 y - 4 = 0$
(4) $4 x + y - 3 = 0$

If the common tangents to the parabola, $x ^ { 2 } = 4 y$ and the circle, $x ^ { 2 } + y ^ { 2 } = 4$ intersect at the point $P$, then the distance of $P$ from the origin (units), is:
(1) $2 ( 3 + 2 \sqrt { 2 } )$
(2) $3 + 2 \sqrt { 2 }$
(3) $\sqrt { 2 } + 1$
(4) $2 ( \sqrt { 2 } + 1 )$

If the tangent at $( 1,7 )$ to the curve $x ^ { 2 } = y - 6$ touch the circle $x ^ { 2 } + y ^ { 2 } + 16 x + 12 y + c = 0$ then the value of $c$ is:
(1) 95
(2) 195
(3) 185
(4) 85

Two parabolas with a common vertex and with axes along the $x$-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) $x + 2 y + 3 = 0$
(4) $4 ( x + y ) + 3 = 0$

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) $3 + 4 \sqrt { 2 }$