Circle(s) touching $x$-axis at a distance 3 from the origin and having an intercept of length $2 \sqrt { 7 }$ on $y$-axis is (are)
(A) $x ^ { 2 } + y ^ { 2 } - 6 x + 8 y + 9 = 0$
(B) $x ^ { 2 } + y ^ { 2 } - 6 x + 7 y + 9 = 0$
(C) $x ^ { 2 } + y ^ { 2 } - 6 x - 8 y + 9 = 0$
(D) $x ^ { 2 } + y ^ { 2 } - 6 x - 7 y + 9 = 0$

A circle $S$ passes through the point $(0,1)$ and is orthogonal to the circles $(x-1)^2 + y^2 = 16$ and $x^2 + y^2 = 1$. Then
(A) radius of $S$ is 8
(B) radius of $S$ is 7
(C) centre of $S$ is $(-7, 1)$
(D) centre of $S$ is $(-8, 1)$

Let $P$ and $Q$ be distinct points on the parabola $y ^ { 2 } = 2 x$ such that a circle with $P Q$ as diameter passes through the vertex $O$ of the parabola. If $P$ lies in the first quadrant and the area of the triangle $\triangle O P Q$ is $3 \sqrt { 2 }$, then which of the following is (are) the coordinates of $P$?
(A) $( 4,2 \sqrt { 2 } )$
(B) $( 9,3 \sqrt { 2 } )$
(C) $\left( \frac { 1 } { 4 } , \frac { 1 } { \sqrt { 2 } } \right)$
(D) $( 1 , \sqrt { 2 } )$

Let $O$ be the centre of the circle $x^{2} + y^{2} = r^{2}$, where $r > \frac{\sqrt{5}}{2}$. Suppose $PQ$ is a chord of this circle and the equation of the line passing through $P$ and $Q$ is $2x + 4y = 5$. If the centre of the circumcircle of the triangle $OPQ$ lies on the line $x + 2y = 4$, then the value of $r$ is $\_\_\_\_$

Consider a triangle $\Delta$ whose two sides lie on the x-axis and the line $x + y + 1 = 0$. If the orthocenter of $\Delta$ is $(1,1)$, then the equation of the circle passing through the vertices of the triangle $\Delta$ is
(A) $x^2 + y^2 - 3x + y = 0$
(B) $x^2 + y^2 + x + 3y = 0$
(C) $x^2 + y^2 + 2y - 1 = 0$
(D) $x^2 + y^2 + x + y = 0$

The equation of the circle passing through the point $( 1,2 )$ and through the points of intersection of $x ^ { 2 } + y ^ { 2 } - 4 x - 6 y - 21 = 0$ and $3 x + 4 y + 5 = 0$ is given by
(1) $x ^ { 2 } + y ^ { 2 } + 2 x + 2 y + 11 = 0$
(2) $x ^ { 2 } + y ^ { 2 } - 2 x + 2 y - 7 = 0$
(3) $x ^ { 2 } + y ^ { 2 } + 2 x - 2 y - 3 = 0$
(4) $x ^ { 2 } + y ^ { 2 } + 2 x + 2 y - 11 = 0$

The equation of the circle passing through the point $(1,0)$ and $(0,1)$ and having the smallest radius is
(1) $x^{2}+y^{2}-2x-2y+1=0$
(2) $x^{2}+y^{2}+2x+2y-7=0$
(3) $x^{2}+y^{2}-x-y=0$
(4) $x^{2}+y^{2}+x+y-2=0$

The length of the diameter of the circle which touches the $x$-axis at the point $(1,0)$ and passes through the point $(2,3)$ is
(1) $\frac{10}{3}$
(2) $\frac{3}{5}$
(3) $\frac{6}{5}$
(4) $\frac{5}{3}$

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 $b \in ( - \infty , 1 )$

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.

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) $(2, -5)$

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) $x ^ { 2 } + y ^ { 2 } + 3 x + y - 2 = 0$

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) $x^2 + y^2 - 4x + 9y + 18 = 0$

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) $x^2 + y^2 - 4x + 9y + 18 = 0$

A circle passes through the points $( 2,3 )$ and $( 4,5 )$. If its centre lies on the line $y - 4 x + 3 = 0$, then its radius is equal to :
(1) $\sqrt { 5 }$
(2) $\sqrt { 2 }$
(3) 2
(4) 1

A circle passes through the points $( 2,3 )$ and $( 4,5 )$. If its centre lies on the line, $y - 4 x + 3 = 0$, then its radius is equal to
(1) $\sqrt { 5 }$
(2) 1
(3) $\sqrt { 2 }$
(4) 2

The circle passing through the intersection of the circles, $x ^ { 2 } + y ^ { 2 } - 6 x = 0$ and $x ^ { 2 } + y ^ { 2 } - 4 y = 0$ having its centre on the line, $2 x - 3 y + 12 = 0$, also passes through the point:
(1) $( - 1,3 )$
(2) $( - 3,6 )$
(3) $( - 3,1 )$
(4) $( 1 , - 3 )$

The centre of the circle passing through the point $(0,1)$ and touching the parabola $y=x^{2}$ at the point $(2,4)$ is
(1) $\left(\frac{-53}{10},\frac{16}{5}\right)$
(2) $\left(\frac{6}{5},\frac{53}{10}\right)$
(3) $\left(\frac{3}{10},\frac{16}{5}\right)$
(4) $\left(\frac{-16}{5},\frac{53}{10}\right)$

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) $\sqrt { 10 }$

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

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) $4 ( S + R )$