Circles and its Equations

Two circles touch each other internally. The radii of these circles are $2 \mathrm{cm}$  and $3 \mathrm{cm}$ respectively. Find the length of the largest chord of the outer circle which touches the internal circle at a point?$$

Tangents drawn from the point $P(1, 8)$ to the circle 

$x^2+y^2-6x-4y-11=0$

touch the circle at the points $A$ and $B$. The equation of the circumcircle of the triangle $PAB$ is

Extremities of a diagonal of a rectangle are $\left(0, 0\right)$ and $\left(4, 3\right).$ Find the equation of the tangents to the circumcircle of a rectangle which are parallel to this diagonal.

The circle $C: x^2+y^2+kx+\left(1+k\right)y-\left(k+1\right)=0$ passes through two fixed points for every real number $K$. The minimum value of the radius of circle $C$ is

The radius of the circle, having centre at (2, 1), whose one of the chord is a diameter of the circle ${\text{x}}^2+{\text{y}}^2-2\text{x}-6\text{y}+6=0$

If two diameters of a circle of circumference $10\pi$ lie along the lines $2x+3y+1=0$ and $3x-y-4=0,$ then the equation of circle is

If $(4, 7)$ and $(-2, -1)$ are ends of a diameter of a circle which intersects $X$-axis at $A$ and $B,$ then $AB$ is equal to $\_\_\_\_\_\_\_\_\_$

The equation of a circle with centre on $y-$axis and passing through the origin and the point $(3, 4)$ is

The number of integer values of $k$ for which the equation $x^2+y^2+\left(k-1\right)x-ky+5=0$ represents a circle whose radius cannot exceed $3$, is

Let $C$ be the circle whose diameter is the line segment formed by the line $3x+2y=6$ intercepted by the coordinate axes. Then $C$ also passes through the point.

Consider a family of circles which are passing through the point $\left(-1,1\right)$ and are tangent to  $x-$axis. If $\left(h,k\right)$ are the coordinate of the centre of the circles, then the set of values of $k$ is given by the interval

$\left(1,2\sqrt{2}\right)$ is a point on circle $x^2+y^2=9$. Which of the following is a point on the circle at $2$ units distance from $\left(1,2\sqrt{2}\right)$?

If the abscissa and ordinates of two points $P$ and $Q$ are roots of the equations $x^2+2ax-b^2=0$ and $y^2+2py-q^2=0$ respectivaly, then the equation of the circle with $PQ$ as diameter, is

The other end of the diameter through the point $\left(-1, 1\right)$ on the circle $x^2+y^2-6x+4y-12=0$ is

The area of the circle $x^2-2x+y^2-10y+k=0$ is $25\pi .$ The value of $k$ is equal to

Centre of circle whose normals are $x^2-2xy-3x+6y=0$, is

Let $A\left(1,2\right), B\left(3,4\right)$ be two points and $C\left(x,y\right)$ be a point such that area of $∆ABC$ is $3$ sq. units and $\left(x-1\right)\left(x-3\right)+\left(y-2\right)\left(y-4\right)=0$. Then number of positions of $C$, in the$xy$ plane, is 

The equation of the circle concentric with the circle $x^2+y^2-6x-4y-12=0$ and touching $Y$-axis

If the circles $x^2+y^2=9$ and $x^2+y^2+2\alpha x+2y+1=0$ touch each other internally, then $\alpha$ is equal to