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

If two tangents drawn from a point $P$ to the parabola $y ^ { 2 } = 16 ( x - 3 )$ are at right angles, then the locus of point $P$ is: (1) $x + 4 = 0$ (2) $x + 2 = 0$ (3) $x + 3 = 0$ (4) $x + 1 = 0$

Let $A ( 1,4 )$ and $B ( 1 , - 5 )$ be two points. Let $P$ be a point on the circle $( ( x - 1 ) ) ^ { 2 } + ( y - 1 ) ^ { 2 } = 1$, such that $( P A ) ^ { 2 } + ( P B ) ^ { 2 }$ have maximum value, then the points, $P , A$ and $B$ lie on
(1) a hyperbola
(2) a straight line
(3) an ellipse
(4) a parabola

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 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) $4\sqrt{5}$

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) $a b = \frac { c + d } { a + b }$

Choose the correct statement about two circles whose equations are given below: $x ^ { 2 } + y ^ { 2 } - 10 x - 10 y + 41 = 0$ $x ^ { 2 } + y ^ { 2 } - 22 x - 10 y + 137 = 0$
(1) circles have same centre
(2) circles have no meeting point

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 $\_\_\_\_$.

Consider a triangle having vertices $A ( - 2,3 ) , B ( 1,9 )$ and $C ( 3,8 )$. If a line $L$ passing through the circumcentre of triangle $ABC$, bisects line $BC$, and intersects $y$-axis at point $\left( 0 , \frac { \alpha } { 2 } \right)$, then the value of real number $\alpha$ is $\underline{\hspace{1cm}}$.

If the variable line $3 x + 4 y = \alpha$ lies between the two circles $( x - 1 ) ^ { 2 } + ( y - 1 ) ^ { 2 } = 1$ and $( x - 9 ) ^ { 2 } + ( y - 1 ) ^ { 2 } = 4$, without intercepting a chord on either circle, then the sum of all the integral values of $\alpha$ is