Circles practice · Mixed difficulty

Two circles ${x^2} + {y^2} - 2x - 3 = 0$ and ${x^2} + {y^2} - 4x - 6y - 8 = 0$ are such that

If the area of the circle $4{x^2} + 4{y^2} - 8x + 16y + k = 0$ is $9\pi$ sq unit, then the value of $k$ is

If the abscissa and the ordinates of two points $A$ and $B$ are the roots of the equations $x^2+2 a x-b^2=0$ and $y^2+2 p y-q^2=0$ respectively.
Then, the equation of circle with $A B$ as diameter is

Suppose a circle passes through $\left( {2,2} \right)$ and $\left( {9,9} \right)$ and touches the $x$-axis at P. If $O$ is the origin, then $OP$ is equal to

Find the radius of the circle whose centre is in fourth quadrant and which touches as each of the three lines $x=0$, $y=0$ and $3 x+4 y-12=0.$

Find the parametric equations of the circle $x^2+y^2-4 x-2 y+$ $1=0$.

The equation of the tangent to the circle ${x^2} + {y^2} = 4$, which are parallel to $x + 2y + 3 = 0$, are

The distinct points $A\left( {0,\;0} \right),\;B\left( {0,\;1} \right),\;C\left( {1,\;0} \right)$ and $D\left( {2a,\;3a} \right)$ are concyclic, then

If the equation of the circle is $x^2+y^2-8 x+10 y-12=0$, then

I. Centre of the circle is $(4,-5)$.

II. Radius of the circle is $\sqrt{53}$.

Let a circle $C_{1}$ be obtained on rolling the circle $x^{2}+$ $y^{2}-4 x-6 y+11=0$ upwards 4 units on the tangent $T$ to it at the point $(3,2)$. Let $C_{2}$ be the image of $C_{1}$ in $T$. Let $A$ and $B$ be the centers of circles $C_{1}$ and $C_{2}$ respectively, and $M$ and $N$ be respectively the feet of perpendiculars drawn from $A$ and $B$ on the $x$-axis. Then the area of the trapezium AMNB is:

The locus of a point which moves so that the ratio of the length of the tangents to the circles ${x^2} + {y^2} + 4x + 3 = 0$ and ${x^2} + {y^2} - 6x + 5 = 0$ is $2:3$, is

Discuss the position of the points $(1,2)$ and $(6,0)$ with respect to the circle $x^2+y^2-4 x+2 y-11=0$.

Equation of a circle with centre $C(h, k)$ and radius 5 such that $h^2-3 h+2=0$ and $k^2+5 k-6=0$ is

The area of the circle whose centre is at $\left( {2,3} \right)$ and passing through $\left( {4,6} \right)$, is

The locus of the centres of the circles for which one end of a diameter is $(1,1)$ while the other end is on the line $x+y=3$, is

The line $3x - 2y = k$ meets the circle ${x^2} + {y^2} = 4{r^2}$ at only one point if ${k^2}$ is equal to

The equation of the circle which touches the axes of the coordinates and the line $\frac{x}{3} + \frac{y}{4} = 1$ and whose centre lies in the first quadrant is ${x^2} + {y^2} - 2cx - 2cy + {c^2} = 0,$ where $c$ is

Find the equation of the circle which passes through the point of intersection of the lines $3 x-2 y-1=0$ and $4 x+y-27=0$ and whose centre is $(2,-3)$.

The intercept on the line $y=x$ by the circle $x^2+y^2-2 x=0$ is $A B$. The equation of the circle with $A B$ as a diameter, is

If the straight line $a x+b y=2 ; a, b \neq 0$ touches the circle $x^2+y^2-2 x=3$ and is normal to circle $x^2+y^2-4 y=6$, then the values of $a$ and $b$ are

The equation of a circle $C$ is ${x^2} + {y^2} - 6c - 8y - 11 = 0.$ The number of real points at which the circle drawn with points $\left( {1,8} \right)$ and $\left( {0,0} \right)$ as the ends of a diameter cuts the circle, $C$ is

If the line $y=m x-(m-1)$ cuts the circle $x^2+y^2=4$ at two real and distinct points, then

Two parallel chords of a circle $S$ have length 10 and 14 and are 6 units apart. If a regular polygon of 12 sides is inscribed in a circle $S$, then find the area of regular polygon.

If $a > 2b > 0$, then the positive value of $m$ for which $y = mx - b\sqrt {1 + {m^2}}$ is a common tangent to ${x^2} + {y^2} = {b^2}$ and ${\left( {x - a} \right)^2} + {y^2} = {b^2}$, is

The circle $x^2+y^{2}=4 x+8 y+5$ intersects the line $3 x-4 y=m$ at two distinct points if

The number of common tangents to the two circles ${x^2} + {y^2} - 8x + 2y = 0$ and ${x^2} + {y^2} - 2x - 16y + 25 = 0$ is

The equation of the circle with centre on the $x$-axis and touching the line $3 x+4 y-11=0$ at point $(1,2)$ is

A circle has two of its diameters along the lines $2 x+$ $3 y-18=0$ and $3 x-y-5=0$ and touches the line $x+2 y+4=0$. Equation of the circle is

If $5 x-12 y+10=0$ and $12 y-5 x+16=0$ are two tangents to a circle, then the radius the circle, is

The equation of the circle which passes through the intersection of ${x^2} + {y^2} + 13x - 3y = 0$ and $2{x^2} + 2{y^2} + 4x - 7y - 25 = 0$ and whose centre lies on $13x + 30y = 0$, is