Consider a circle of radius 6 as given in the diagram below. Let $B$, $C , D$ and $E$ be points on the circle such that $B D$ and $C E$, when extended, intersect at $A$. If $A D$ and $A E$ have length 5 and 4 respectively, and $D B C$ is a right angle, then show that the length of $B C$ is $\frac { 12 + 9 \sqrt { 15 } } { 5 }$.

Let $C$ be a circle of area $A$ with centre at $O$. Let $P$ be a moving point such that its distance from $O$ is always twice the length of a tangent drawn from $P$ to the circle. Then the point $P$ must move along
(A) the sides of a square centred at $O$, with area $\frac{4}{3}A$.
(B) the sides of an equilateral triangle centred at $O$, with area $4A$.
(C) a circle centred at $O$, with area $\frac{4}{3}A$.
(D) a circle centred at $O$, with area $4A$.

Let $P = \left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)$ and $Q = \left(-\frac{1}{\sqrt{2}}, -\frac{1}{\sqrt{2}}\right)$ be two vertices of a regular polygon having 12 sides such that $PQ$ is a diameter of the circle circumscribing the polygon. Which of the following points is not a vertex of this polygon?
(A) $\left(\frac{\sqrt{3}-1}{2\sqrt{2}}, \frac{\sqrt{3}+1}{2\sqrt{2}}\right)$
(B) $\left(\frac{\sqrt{3}+1}{2\sqrt{2}}, \frac{\sqrt{3}-1}{2\sqrt{2}}\right)$
(C) $\left(\frac{\sqrt{3}+1}{2\sqrt{2}}, \frac{1-\sqrt{3}}{2\sqrt{2}}\right)$
(D) $\left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$.

Suppose that $P Q$ and $R S$ are two chords of a circle intersecting at a point $O$. It is given that $P O = 3 \mathrm {~cm}$ and $S O = 4 \mathrm {~cm}$. Moreover, the area of the triangle $P O R$ is $7 \mathrm {~cm} ^ { 2 }$. Find the area of the triangle $Q O S$.

In the following picture, $A B C$ is an isosceles triangle with an inscribed circle with center $O$. Let $P$ be the mid-point of $B C$. If $A B = A C = 15$ and $B C = 10$, then $O P$ equals:
(A) $\frac { \sqrt { 5 } } { \sqrt { 2 } }$
(B) $\frac { 5 } { \sqrt { 2 } }$
(C) $2 \sqrt { 5 }$
(D) $5 \sqrt { 2 }$.

Chords $A B$ and $C D$ of a circle intersect at right angle at the point $P$. If the lengths of $A P , P B , C P , P D$ are $2,6,3,4$ units respectively, then the radius of the circle is:
(A) 4
(B) $\frac { \sqrt { 65 } } { 2 }$
(C) $\frac { \sqrt { 66 } } { 2 }$
(D) $\frac { \sqrt { 67 } } { 2 }$

Two vertices of a square lie on a circle of radius $r$ and the other two vertices lie on a tangent to this circle. Then the length of the side of the square is
(A) $\frac { 3 r } { 2 }$
(B) $\frac { 4 r } { 3 }$
(C) $\frac { 6 r } { 5 }$
(D) $\frac { 8 r } { 5 }$.

Consider the curves $x ^ { 2 } + y ^ { 2 } - 4 x - 6 y - 12 = 0,9 x ^ { 2 } + 4 y ^ { 2 } - 900 = 0$ and $y ^ { 2 } - 6 y - 6 x + 51 = 0$. The maximum number of disjoint regions into which these curves divide the $XY$-plane (excluding the curves themselves), is
(A) 4 .
(B) 5 .
(C) 6 .
(D) 7 .

If two real numbers $x$ and $y$ satisfy $( x + 5 ) ^ { 2 } + ( y - 10 ) ^ { 2 } = 196$, then the minimum possible value of $x ^ { 2 } + 2 x + y ^ { 2 } - 4 y$ is
(A) $271 - 112 \sqrt { 5 }$.
(B) $14 - 4 \sqrt { 5 }$.
(C) $276 - 112 \sqrt { 5 }$.
(D) $9 - 4 \sqrt { 5 }$.

Consider a triangle with vertices $( 0,0 ) , ( 1,2 )$ and $( - 4,2 )$. Let $A$ be the area of the triangle and $B$ be the area of the circumcircle of the triangle. Then $\frac { B } { A }$ equals
(A) $\frac { \pi } { 2 }$.
(B) $\frac { 5 \pi } { 4 }$.
(C) $\frac { 3 } { \sqrt { 2 } } \pi$.
(D) $2 \pi$.

For real numbers $a , b , c , d , a ^ { \prime } , b ^ { \prime } , c ^ { \prime } , d ^ { \prime }$, consider the system of equations $$\begin{aligned} a x ^ { 2 } + a y ^ { 2 } + b x + c y + d & = 0 \\ a ^ { \prime } x ^ { 2 } + a ^ { \prime } y ^ { 2 } + b ^ { \prime } x + c ^ { \prime } y + d ^ { \prime } & = 0 \end{aligned}$$ If $S$ denotes the set of all real solutions $( x , y )$ of the above system of equations, then the number of elements in $S$ can never be
(A) 0.
(B) 1.
(C) 2.
(D) 3.

In the following figure, $O A B$ is a quarter-circle. The unshaded region is a circle to which $O A$ and $C D$ are tangents. If $C D$ is of length 10 and is parallel to $O A$, then the area of the shaded region in the above figure equals
(A) $25 \pi$.
(B) $50 \pi$.
(C) $75 \pi$.
(D) $100 \pi$.

The angle subtended at the origin by the common chord of the circles $x^2 + y^2 - 6x - 6y = 0$ and $x^2 + y^2 = 36$ is
(A) $\pi/2$
(B) $\pi/4$
(C) $\pi/3$
(D) $2\pi/3$

Consider points of the form $\left(n, n^k\right)$, where $n$ and $k$ are integers with $n \geq 0$, $k \geq 1$. How many such points are strictly inside the circle of radius 10 with centre at the origin?
(A) 11
(B) 12
(C) 15
(D) 17

In the adjoining figure, $C$ is the centre of the circle drawn, $A, F, E$ lie on the circle and $BCDF$ is a rectangle. If $\frac{DE}{AB} = 2$, then $\frac{FE}{FA}$ equals
(A) $\sqrt{\frac{3}{2}}$
(B) $\sqrt{2}$
(C) $\sqrt{\frac{5}{2}}$
(D) $\sqrt{3}$

Consider a circle with centre $O$. Two chords $A B$ and $C D$ extended intersect at a point $P$ outside the circle. If $\angle A O C = 43 ^ { \circ }$ and $\angle B P D = 18 ^ { \circ }$, then the value of $\angle B O D$ is
(a) $36 ^ { \circ }$.
(B) $29 ^ { \circ }$.
(C) $7 ^ { \circ }$.
(D) $25 ^ { \circ }$.

A triangle $A B C$ has a fixed base $B C$. If $A B : A C = 1 : 2$, then the locus of the vertex $A$ is
(a) a circle whose centre is the midpoint of $B C$.
(b) a circle whose centre is on the line $B C$ but not the midpoint of $B C$.
(c) a straight line.
(d) none of the above.

Consider the two curves $$\begin{aligned} & C _ { 1 } : y ^ { 2 } = 4 x \\ & C _ { 2 } : x ^ { 2 } + y ^ { 2 } - 6 x + 1 = 0 \end{aligned}$$ Then,
(A) $C _ { 1 }$ and $C _ { 2 }$ touch each other only at one point
(B) $C _ { 1 }$ and $C _ { 2 }$ touch each other exactly at two points
(C) $C _ { 1 }$ and $C _ { 2 }$ intersect (but do not touch) at exactly two points
(D) $C _ { 1 }$ and $C _ { 2 }$ neither intersect nor touch each other

Let $a$ and $b$ be non-zero real numbers. Then, the equation $$\left( a x ^ { 2 } + b y ^ { 2 } + c \right) \left( x ^ { 2 } - 5 x y + 6 y ^ { 2 } \right) = 0$$ represents
(A) four straight lines, when $c = 0$ and $a , b$ are of the same sign
(B) two straight lines and a circle, when $a = b$, and $c$ is of sign opposite to that of $a$
(C) two straight lines and a hyperbola, when $a$ and $b$ are of the same sign and $c$ is of sign opposite to that of $a$
(D) a circle and an ellipse, when $a$ and $b$ are of the same sign and $c$ is of sign opposite to that of $a$

A straight line through the vertex $P$ of a triangle $P Q R$ intersects the side $Q R$ at the point $S$ and the circumcircle of the triangle $P Q R$ at the point $T$. If $S$ is not the centre of the circumcircle, then
(A) $\frac { 1 } { P S } + \frac { 1 } { S T } < \frac { 2 } { \sqrt { Q S \times S R } }$
(B) $\frac { 1 } { P S } + \frac { 1 } { S T } > \frac { 2 } { \sqrt { Q S \times S R } }$
(C) $\frac { 1 } { P S } + \frac { 1 } { S T } < \frac { 4 } { Q R }$
(D) $\frac { 1 } { P S } + \frac { 1 } { S T } > \frac { 4 } { Q R }$

Consider
$$\begin{aligned} & L _ { 1 } : 2 x + 3 y + p - 3 = 0 \\ & L _ { 2 } : 2 x + 3 y + p + 3 = 0 \end{aligned}$$
where $p$ is a real number, and $C : x ^ { 2 } + y ^ { 2 } + 6 x - 10 y + 30 = 0$. STATEMENT-1 : If line $L _ { 1 }$ is a chord of circle $C$, then line $L _ { 2 }$ is not always a diameter of circle $C$.
and
STATEMENT-2 : If line $L _ { 1 }$ is a diameter of circle $C$, then line $L _ { 2 }$ is not a chord of circle $C$.
(A) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1
(B) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1
(C) STATEMENT-1 is True, STATEMENT-2 is False
(D) STATEMENT-1 is False, STATEMENT-2 is True

A circle $C$ of radius 1 is inscribed in an equilateral triangle $P Q R$. The points of contact of $C$ with the sides $P Q , Q R , R P$ are $D , E , F$, respectively. The line $P Q$ is given by the equation $\sqrt { 3 } x + y - 6 = 0$ and the point $D$ is $\left( \frac { 3 \sqrt { 3 } } { 2 } , \frac { 3 } { 2 } \right)$. Further, it is given that the origin and the centre of $C$ are on the same side of the line $P Q$.
The equation of circle $C$ is
(A) $\quad ( x - 2 \sqrt { 3 } ) ^ { 2 } + ( y - 1 ) ^ { 2 } = 1$
(B) $( x - 2 \sqrt { 3 } ) ^ { 2 } + \left( y + \frac { 1 } { 2 } \right) ^ { 2 } = 1$
(C) $\quad ( x - \sqrt { 3 } ) ^ { 2 } + ( y + 1 ) ^ { 2 } = 1$
(D) $( x - \sqrt { 3 } ) ^ { 2 } + ( y - 1 ) ^ { 2 } = 1$

Tangents drawn from the point $P ( 1,8 )$ to the circle
$$x ^ { 2 } + y ^ { 2 } - 6 x - 4 y - 11 = 0$$
touch the circle at the points $A$ and $B$. The equation of the circumcircle of the triangle $P A B$ is
(A) $x ^ { 2 } + y ^ { 2 } + 4 x - 6 y + 19 = 0$
(B) $x ^ { 2 } + y ^ { 2 } - 4 x - 10 y + 19 = 0$
(C) $x ^ { 2 } + y ^ { 2 } - 2 x + 6 y - 29 = 0$
(D) $x ^ { 2 } + y ^ { 2 } - 6 x - 4 y + 19 = 0$

The centres of two circles $C_{1}$ and $C_{2}$ each of unit radius are at a distance of 6 units from each other. Let $P$ be the mid point of the line segment joining the centres of $C_{1}$ and $C_{2}$ and $C$ be a circle touching circles $C_{1}$ and $C_{2}$ externally. If a common tangent to $C_{1}$ and $C$ passing through $P$ is also a common tangent to $C_{2}$ and $C$, then the radius of the circle $C$ is

Tangents are drawn from the point $P ( 3,4 )$ to the ellipse $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 4 } = 1$ touching the ellipse at points A and B.
The equation of the locus of the point whose distances from the point $P$ and the line AB are equal, is
A) $9 x ^ { 2 } + y ^ { 2 } - 6 x y - 54 x - 62 y + 241 = 0$
B) $x ^ { 2 } + 9 y ^ { 2 } + 6 x y - 54 x + 62 y - 241 = 0$
C) $9 x ^ { 2 } + 9 y ^ { 2 } - 6 x y - 54 x - 62 y - 241 = 0$
D) $x ^ { 2 } + y ^ { 2 } - 2 x y + 27 x + 31 y - 120 = 0$