If the circle $x ^ { 2 } + y ^ { 2 } - 6 x - 8 y + \left( 25 - a ^ { 2 } \right) = 0$ touches the axis of $x$, then $a$ equals.
(1) 0
(2) $\pm 4$
(3) $\pm 2$
(4) $\pm 3$

If each of the lines $5 x + 8 y = 13$ and $4 x - y = 3$ contains a diameter of the circle $x ^ { 2 } + y ^ { 2 } - 2 \left( a ^ { 2 } - 7 a + 11 \right) x - 2 \left( a ^ { 2 } - 6 a + 6 \right) y + b ^ { 3 } + 1 = 0$, then :
(1) $a = 5$ and $b \notin ( - 1,1 )$
(2) $a = 1$ and $b \notin ( - 1,1 )$
(3) $a = 2$ and $b \notin ( - \infty , 1 )$
(4) $a = 5$ and $b \in ( - \infty , 1 )$

The point of intersection of the normals to the parabola $y ^ { 2 } = 4 x$ at the ends of its latus rectum is :
(1) $( 0,2 )$
(2) $( 3,0 )$
(3) $( 0,3 )$
(4) $( 2,0 )$

Statement 1: The only circle having radius $\sqrt { 10 }$ and a diameter along line $2x + y = 5$ is $x ^ { 2 } + y ^ { 2 } - 6x + 2y = 0$. Statement 2: $2x + y = 5$ is a normal to the circle $x ^ { 2 } + y ^ { 2 } - 6x + 2y = 0$.
(1) Statement 1 is false; Statement 2 is true.
(2) Statement 1 is true; Statement 2 is true, Statement 2 is a correct explanation for Statement 1.
(3) Statement 1 is true; Statement 2 is false.
(4) Statement 1 is true; Statement 2 is true, Statement 2 is not a correct explanation for Statement 1.

A tangent to the hyperbola $\frac { x ^ { 2 } } { 4 } - \frac { y ^ { 2 } } { 2 } = 1$ meets $x$-axis at P and $y$-axis at Q. Lines PR and QR are drawn such that OPRQ is a rectangle (where O is the origin). Then R lies on :
(1) $\frac { 4 } { x ^ { 2 } } + \frac { 2 } { y ^ { 2 } } = 1$
(2) $\frac { 2 } { x ^ { 2 } } - \frac { 4 } { y ^ { 2 } } = 1$
(3) $\frac { 2 } { x ^ { 2 } } + \frac { 4 } { y ^ { 2 } } = 1$
(4) $\frac { 4 } { x ^ { 2 } } - \frac { 2 } { y ^ { 2 } } = 1$

The circle passing through $(1, -2)$ and touching the axis of $x$ at $(3, 0)$ also passes through the point
(1) $(5, -2)$
(2) $(-2, 5)$
(3) $(-5, 2)$
(4) $(2, -5)$

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 equation of the circle passing through the foci of the ellipse $\frac{x^2}{16} + \frac{y^2}{9} = 1$, and having centre at $(0, 3)$ is
(1) $x^2 + y^2 - 6y - 5 = 0$
(2) $x^2 + y^2 - 6y + 5 = 0$
(3) $x^2 + y^2 - 6y - 7 = 0$
(4) $x^2 + y^2 - 6y + 7 = 0$

The equation of the circle described on the chord $3 x + y + 5 = 0$ of the circle $x ^ { 2 } + y ^ { 2 } = 16$ as the diameter is
(1) $x ^ { 2 } + y ^ { 2 } + 3 x + y + 1 = 0$
(2) $x ^ { 2 } + y ^ { 2 } + 3 x + y - 22 = 0$
(3) $x ^ { 2 } + y ^ { 2 } + 3 x + y - 11 = 0$
(4) $x ^ { 2 } + y ^ { 2 } + 3 x + y - 2 = 0$

Let $C$ be the circle with center at $( 1,1 )$ and radius $= 1$. If $T$ is the circle centered at $( 0 , y )$, passing through the origin and touching the circle $C$ externally, then the radius of $T$ is equal to
(1) $\frac { 1 } { 2 }$
(2) $\frac { 1 } { 4 }$
(3) $\frac { \sqrt { 3 } } { \sqrt { 2 } }$
(4) $\frac { \sqrt { 3 } } { 2 }$

The locus of the foot of perpendicular drawn from the centre of the ellipse $x ^ { 2 } + 3 y ^ { 2 } = 6$ on any tangent to it is
(1) $\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = 6 x ^ { 2 } + 2 y ^ { 2 }$
(2) $\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = 6 x ^ { 2 } - 2 y ^ { 2 }$
(3) $\left( x ^ { 2 } - y ^ { 2 } \right) ^ { 2 } = 6 x ^ { 2 } + 2 y ^ { 2 }$
(4) $\left( x ^ { 2 } - y ^ { 2 } \right) ^ { 2 } = 6 x ^ { 2 } - 2 y ^ { 2 }$

Let $a$ and $b$ be any two numbers satisfying $\frac { 1 } { a ^ { 2 } } + \frac { 1 } { b ^ { 2 } } = \frac { 1 } { 4 }$. Then, the foot of perpendicular from the origin on the variable line $\frac { x } { a } + \frac { y } { b } = 1$ lies on:
(1) A circle of radius $= 2$
(2) A hyperbola with each semi-axis $= \sqrt { 2 }$.
(3) A hyperbola with each semi-axis $= 2$
(4) A circle of radius $= \sqrt { 2 }$

If the point $( 1,4 )$ lies inside the circle $x ^ { 2 } + y ^ { 2 } - 6 x + 10 y + p = 0$ and the circle does not touch or intersect the coordinate axes, then the set of all possible values of $p$ is the interval
(1) $( 25,39 )$
(2) $( 25,29 )$
(3) $( 0,25 )$
(4) $( 9,25 )$

The slope of the line touching both the parabolas $y ^ { 2 } = 4 x$ and $x ^ { 2 } = - 32 y$ is
(1) $\frac { 1 } { 8 }$
(2) $\frac { 2 } { 3 }$
(3) $\frac { 1 } { 2 }$
(4) $\frac { 3 } { 2 }$

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

Let $O$ be the vertex and $Q$ be any point on the parabola, $x ^ { 2 } = 8 y$. If the point $P$ divides the line segment $OQ$ internally in the ratio $1 : 3$, then the locus of $P$ is
(1) $x ^ { 2 } = 2 y$
(2) $x ^ { 2 } = y$
(3) $y ^ { 2 } = x$
(4) $y ^ { 2 } = 2 x$

The area (in sq. units) of the quadrilateral formed by the tangents at the end points of the latus rectum to the ellipse $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 5 } = 1$, is
(1) 27
(2) $\frac { 27 } { 4 }$
(3) 18
(4) $\frac { 27 } { 2 }$

Locus of the image of the point $(2, 3)$ in the line $(2x - 3y + 4) + k(x - 2y + 3) = 0$, $k \in \mathbb{R}$, is a:
(1) straight line parallel to $x$-axis
(2) straight line parallel to $y$-axis
(3) circle of radius $\sqrt{2}$
(4) circle of radius $\sqrt{3}$

The eccentricity of an ellipse whose centre is at the origin is $\frac{1}{2}$. If one of its directrices is $x = -4$, then the equation of the normal to it at $\left(1, \frac{3}{2}\right)$ is:
(1) $4x - 2y = 1$
(2) $4x + 2y = 7$
(3) $x + 2y = 4$
(4) $2y - x = 2$

Let $P$ be the point on the parabola, $y^2 = 8x$ which is at a minimum distance from the centre $C$ of the circle, $x^2 + (y+6)^2 = 1$. Then the equation of the circle, passing through $C$ and having its centre at $P$ is: (1) $x^2 + y^2 - 4x + 8y + 12 = 0$ (2) $x^2 + y^2 - x + 4y - 12 = 0$ (3) $x^2 + y^2 - \frac{x}{4} + 2y - 24 = 0$ (4) $x^2 + y^2 - 4x + 9y + 18 = 0$

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) $4 x + y - 3 = 0$

The centres of those circles which touch the circle, $x^2 + y^2 - 8x - 8y - 4 = 0$, externally and also touch the $x$-axis, lie on:
(1) a circle
(2) an ellipse which is not a circle
(3) a hyperbola
(4) a parabola

If one of the diameters of the circle, given by the equation, $x^2 + y^2 - 4x + 6y - 12 = 0$, is a chord of a circle $S$, whose centre is at $(-3, 2)$, then the radius of $S$ is:
(1) $5\sqrt{2}$
(2) $5\sqrt{3}$
(3) $5$
(4) $10$

Let $P$ be the point on the parabola, $y^2 = 8x$, which is at a minimum distance from the centre $C$ of the circle, $x^2 + (y+6)^2 = 1$. Then the equation of the circle, passing through $C$ and having its centre at $P$ is:
(1) $x^2 + y^2 - 4x + 8y + 12 = 0$
(2) $x^2 + y^2 - x + 4y - 12 = 0$
(3) $x^2 + y^2 - \frac{x}{4} + 2y - 24 = 0$
(4) $x^2 + y^2 - 4x + 9y + 18 = 0$