Circle - Coordinate Geometry - Mathematics Previous Year Questions

The equation of the transverse common tangent of the circles $x^2+y^2-6 x-8 y+9=0$ and $x^2+y^2+2 x-2 y+1=0$

From a point $A(0,3)$ on the circle $(x+2)^2+(y-3)^2=4$, a chord $A B$ is drawn and it is extended to a point $Q$ such that $A Q=2 A B$. Then, the locus of $Q$ is

If $m_1, m_2$ are the slopes of the tangents drawn from a point $(1,-3)$ to the circle $x^2+y^2-6 x+4 y+12=0$, then $9\left(m_1^2+m_2^2\right)=$

If $A, B$ are the points of contact of the tangents drawn from the point $P(-2,-3)$ to the circle $x^2+y^2-8 x-10 y+5=0$ and the chord $A B$ subtends an angle $\theta$ at $P$, then $\tan \theta=$

If $\left(0, \frac{3}{4}\right)$ is the radical centre of the circles $S \equiv x^2+y^2+\alpha x+6 y=0, S \equiv x^2+y^2+2 \alpha x+\alpha y+6=0$ and $S^{\prime \prime} \equiv x^2+y^2+6 \alpha x-\alpha y+3=0$, then the distance between the radical centre and the centre of the circle $S^{\prime}=0$ is

If $\theta$ is the angle between the circles

$x^2+y^2-2 x-4 y-4=0$ and $x^2+y^2-8 x-12 y+43=0$, then $|7 \sec \theta-18 \cos \theta|=$

A circle passes through the points $(1,2)$, $(3,4)$. If its centre lines on the line $x-y+3=0$, then its radius is equal to

Let the slope of a diameter $A C$ of a circle of radius 25 units be $\frac{3}{4}$. If $(3,2)$ is the centre of the circle, $A=\left(x_1, y_1\right)$ and $C=\left(x_2, y_2\right)$, then $\frac{x_1 x_2}{y_1 y_2}=$

A line drawn through the point $A(5,7)$ cut the circle $x^2+y^2-36=0$ at the points $P$ and $Q$. Then, $A P \cdot A Q=$

If a circle $C$ passing through $(4,0)$ touches the circle $x^2+y^2+4 x-6 y-12=0$ externally at the point $(1$, -1 ), then the radius of $C$ is

Let $P$ be any point on the circle $x^2+y^2-2 x-1=0$ and $C$ be its centre. Let $A B$ be the chord of contact of $P$ with respect to the circle $x^2+y^2-2 x=0$. Then, the locus of the circumcentre of the $\triangle C A B$ is

If the common chord of the circles $x^2+y^2+4 y=0$ and $x^2+y^2-4 x-5=0$ is the diameter of the circle $S=0$, then the abscissa of the centre of the circle $S=0$ is

If the circles $C_1: x^2+y^2+2 x+4 y-20=0$, $C_2: x^2+y^2+6 x-8 y+9=0$ have $n$ common tangents and the length of the tangent drawn from the centre of similitude to the circle $C_2$ is $l$, then $\frac{l}{n^2}=$

If a circle passing through $(1,-2)$ has $x-y=2$ and $2 x+3 y=14$ as its diameters, then the radius of the circle is

The pole of the straight line $9 x+y-28=0$ with respect to the circle $2 x^2+2 y^2-3 x+5 y-7=0$ is

If the angle between the circles

$$x^2+y^2-2 x-4 y+c=0 \text { and } x^2+y^2-4 x-2 y+4=0$$

is $60^{\circ}$, then $c=$

The equation of the circle whose diameter is the common chord of the circles $x^2+y^2+2 x+3 y+1=0$ and $x^2+y^2+4 x+3 y+2=0$ is

The equation of the line perpendicular to the radical axis of two circles $x^2+y^2-5 x+6 y+12=0$, $x^2+y^2+6 x-4 y-14=0$ and passing through $(1,1)$ is

Let the circle $S \equiv x^2+y^2+2 g x+2 f y+c=0$ touch the positive $X$-axis and the positive $Y$-axis. Let $(2,4)$ be a point on the circle $S=0$. If two such circles exist, then the difference of their areas is