If the locus of the mid-point of the line segment from the point $( 3,2 )$ to a point on the circle, $x ^ { 2 } + y ^ { 2 } = 1$ is a circle of radius $r$, then $r$ is equal to
(1) $\frac { 1 } { 4 }$
(2) 1
(3) $\frac { 1 } { 3 }$
(4) $\frac { 1 } { 2 }$

Let the lengths of intercepts on $x$-axis and $y$-axis made by the circle $x ^ { 2 } + y ^ { 2 } + ax + 2ay + c = 0 , ( a < 0 )$ be $2 \sqrt { 2 }$ and $2 \sqrt { 5 }$, respectively. Then the shortest distance from origin to a tangent to this circle which is perpendicular to the line $x + 2y = 0$, is equal to :
(1) $\sqrt { 11 }$
(2) $\sqrt { 7 }$
(3) $\sqrt { 6 }$
(4) $\sqrt { 10 }$

Let the centroid of an equilateral triangle $ABC$ be at the origin. Let one of the sides of the equilateral triangle be along the straight line $x + y = 3$. If $R$ and $r$ be the radius of circumcircle and incircle respectively of $\triangle ABC$, then $( R + r )$ is equal to:
(1) $\frac { 9 } { \sqrt { 2 } }$
(2) $7 \sqrt { 2 }$
(3) $2 \sqrt { 2 }$
(4) $3 \sqrt { 2 }$

Let the circle $S : 36 x ^ { 2 } + 36 y ^ { 2 } - 108 x + 120 y + C = 0$ be such that it neither intersects nor touches the coordinate axes. If the point of intersection of the lines, $x - 2 y = 4$ and $2 x - y = 5$ lies inside the circle $S$, then:
(1) $\frac { 25 } { 9 } < C < \frac { 13 } { 3 }$
(2) $100 < C < 165$
(3) $81 < C < 156$
(4) $100 < C < 156$

Let $r _ { 1 }$ and $r _ { 2 }$ be the radii of the largest and smallest circles, respectively, which pass through the point $( - 4,1 )$ and having their centres on the circumference of the circle $x ^ { 2 } + y ^ { 2 } + 2 x + 4 y - 4 = 0$. If $\frac { r _ { 1 } } { r _ { 2 } } = a + b \sqrt { 2 }$, then $a + b$ is equal to:
(1) 3
(2) 11
(3) 5
(4) 7

Let $C$ be the locus of the mirror image of a point on the parabola $y ^ { 2 } = 4x$ with respect to the line $y = x$. Then the equation of tangent to $C$ at $P ( 2,1 )$ is :
(1) $x - y = 1$
(2) $2x + y = 5$
(3) $x + 3y = 5$
(4) $x + 2y = 4$

In a triangle $PQR$, the co-ordinates of the points $P$ and $Q$ are $(-2, 4)$ and $(4, -2)$ respectively. If the equation of the perpendicular bisector of $PR$ is $2x - y + 2 = 0$, then the centre of the circumcircle of the $\triangle PQR$ is:
(1) $(-1, 0)$
(2) $(-2, -2)$
(3) $(0, 2)$
(4) $(1, 4)$

Let $S _ { 1 } : x ^ { 2 } + y ^ { 2 } = 9$ and $S _ { 2 } : ( x - 2 ) ^ { 2 } + y ^ { 2 } = 1$.
Then the locus of center of a variable circle $S$ which touches $S _ { 1 }$ internally and $S _ { 2 }$ externally always passes through the points:
(1) $( 0 , \pm \sqrt { 3 } )$
(2) $\left( \frac { 1 } { 2 } , \pm \frac { \sqrt { 5 } } { 2 } \right)$
(3) $\left( 2 , \pm \frac { 3 } { 2 } \right)$
(4) $( 1 , \pm 2 )$

If the curve $x ^ { 2 } + 2 y ^ { 2 } = 2$ intersects the line $x + y = 1$ at two points $P$ and $Q$, then the angle subtended by the line segment $PQ$ at the origin is
(1) $\frac { \pi } { 2 } - \tan ^ { - 1 } \left( \frac { 1 } { 3 } \right)$
(2) $\frac { \pi } { 2 } + \tan ^ { - 1 } \left( \frac { 1 } { 3 } \right)$
(3) $\frac { \pi } { 2 } + \tan ^ { - 1 } \left( \frac { 1 } { 4 } \right)$
(4) $\frac { \pi } { 2 } - \tan ^ { - 1 } \left( \frac { 1 } { 4 } \right)$

Two tangents are drawn from a point $P$ to the circle $x ^ { 2 } + y ^ { 2 } - 2 x - 4 y + 4 = 0$, such that the angle between these tangents is $\tan ^ { - 1 } \left( \frac { 12 } { 5 } \right)$, where $\tan ^ { - 1 } \left( \frac { 12 } { 5 } \right) \in ( 0 , \pi )$. If the centre of the circle is denoted by $C$ and these tangents touch the circle at points $A$ and $B$, then the ratio of the areas of $\triangle P A B$ and $\triangle C A B$ is:
(1) $11 : 4$
(2) $9 : 4$
(3) $3 : 1$
(4) $2 : 1$

The length of the latus rectum of a parabola, whose vertex and focus are on the positive $x$-axis at a distance $R$ and $S ( > \mathrm { R } )$ respectively from the origin, is :
(1) $2 ( S - R )$
(2) $2 ( S + R )$
(3) $4 ( S - R )$
(4) $4 ( S + R )$

The locus of the mid-point of the line segment joining the focus of the parabola $y ^ { 2 } = 4 a x$ to a moving point of the parabola, is another parabola whose directrix is:
(1) $x = a$
(2) $x = 0$
(3) $x = - \frac { a } { 2 }$
(4) $x = \frac { a } { 2 }$

The line $2x - y + 1 = 0$ is a tangent to the circle at the point $(2, 5)$ and the centre of the circle lies on $x - 2y = 4$. Then, the radius of the circle is:
(1) $3\sqrt{5}$
(2) $5\sqrt{3}$
(3) $5\sqrt{4}$
(4) $4\sqrt{5}$

Let a tangent be drawn to the ellipse $\frac { x ^ { 2 } } { 27 } + y ^ { 2 } = 1$ at $( 3 \sqrt { 3 } \cos \theta , \sin \theta )$ where $\theta \in \left( 0 , \frac { \pi } { 2 } \right)$. Then the value of $\theta$ such that the sum of intercepts on axes made by this tangent is minimum is equal to:
(1) $\frac { \pi } { 8 }$
(2) $\frac { \pi } { 4 }$
(3) $\frac { \pi } { 6 }$
(4) $\frac { \pi } { 3 }$

Consider the parabola with vertex $\left(\frac { 1 } { 2 } , \frac { 3 } { 4 }\right)$ and the directrix $y = \frac { 1 } { 2 }$. Let P be the point where the parabola meets the line $x = - \frac { 1 } { 2 }$. If the normal to the parabola at P intersects the parabola again at the point Q, then $(PQ) ^ { 2 }$ is equal to :
(1) $\frac { 25 } { 2 }$
(2) $\frac { 75 } { 8 }$
(3) $\frac { 125 } { 16 }$
(4) $\frac { 15 } { 2 }$

The image of the point $( 3,5 )$ in the line $x - y + 1 = 0$, lies on :
(1) $( x - 2 ) ^ { 2 } + ( y - 4 ) ^ { 2 } = 4$
(2) $( x - 4 ) ^ { 2 } + ( y - 4 ) ^ { 2 } = 8$
(3) $( x - 4 ) ^ { 2 } + ( y + 2 ) ^ { 2 } = 16$
(4) $( x - 2 ) ^ { 2 } + ( y - 2 ) ^ { 2 } = 12$

If the three normals drawn to the parabola, $y ^ { 2 } = 2 x$ pass through the point $( a , 0 ) , a \neq 0$, then $a$ must be greater than :
(1) $\frac { 1 } { 2 }$
(2) $- \frac { 1 } { 2 }$
(3) - 1
(4) 1

Let the tangent to the circle $x ^ { 2 } + y ^ { 2 } = 25$ at the point $R ( 3,4 )$ meet $x$-axis and $y$-axis at point $P$ and $Q$, respectively. If $r$ is the radius of the circle passing through the origin $O$ and having centre at the incentre of the triangle $O P Q$, then $r ^ { 2 }$ is equal to
(1) $\frac { 529 } { 64 }$
(2) $\frac { 125 } { 72 }$
(3) $\frac { 625 } { 72 }$
(4) $\frac { 585 } { 66 }$

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) $25 x ^ { 2 } + 12 y ^ { 2 } = 3600$

Choose the incorrect statement about the two circles whose equations are given below: $x ^ { 2 } + y ^ { 2 } - 10 x - 10 y + 41 = 0$ and $x ^ { 2 } + y ^ { 2 } - 16 x - 10 y + 80 = 0$
(1) Distance between two centres is the average of radii of both the circles.
(2) Both circles' centres lie inside region of one another.
(3) Both circles pass through the centre of each other.
(4) Circles have two intersection points.

Let $\theta$ be the acute angle between the tangents to the ellipse $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 1 } = 1$ and the circle $x^2 + y^2 = 3$ at their points of intersection. Then $\tan\theta$ is equal to:
(1) $\frac{4}{\sqrt{3}}$
(2) $\frac{2}{\sqrt{3}}$
(3) $2$
(4) $\frac{5}{2\sqrt{3}}$

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) $( - 6,0 )$

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 :
(1) $\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } - 16 x ^ { 2 } + 9 y ^ { 2 } = 0$
(2) $\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } - 9 x ^ { 2 } + 144 y ^ { 2 } = 0$
(3) $\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } - 9 x ^ { 2 } - 16 y ^ { 2 } = 0$
(4) $\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } - 9 x ^ { 2 } + 16 y ^ { 2 } = 0$

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

If the curves, $\frac { x ^ { 2 } } { a } + \frac { y ^ { 2 } } { b } = 1$ and $\frac { x ^ { 2 } } { c } + \frac { y ^ { 2 } } { d } = 1$ intersect each other at an angle of $90 ^ { \circ }$, then which of the following relations is TRUE?
(1) $a - c = b + d$
(2) $a - b = c - d$
(3) $a + b = c + d$
(4) $a b = \frac { c + d } { a + b }$