Basics of Circle

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

If $A$ and $B$ are fixed points in the plane such that $\frac{PA}{PB}=k$  (constant) for all $P$ on a given circle, then the value of $k$ cannot be equal to:

The equation of a circle with centre at $\left(2,2\right)$ and passes through the point $\left(4,5\right)$ is

The equation of a circle with center at $\left(-2, 3\right)$ and circumference of $4\pi$ units is

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.

The equation of the circle whose centre lies on the line $2x+y=0$ and touching the lines $4x-3y+10=0$ and $4x-3y-30=0$ is

If the equation of the circumcircle of an equilateral triangle is $x^2+y^2-10 x+12 y+45=0,$ the area of the triangle is

If the equation of the incircle of a triangle is $2x^2+2y^2+5x+y-15=0$ then the area of the triangle is

The length of a side of the equilateral triangle inscribed in the circle $x^2+y^2=a^2$ is

The length of the intercept made by the circle $x^2+y^2-6x-2y-7=0$ on the $x-$axis is

The length of the chord parallel to $y-$axis of the circle $x^2+y^2-6x-8y-9=0$ is

The equation of the incircle of the triangle formed by the straight line $4x+3y=6$ with the coordinate axes is

If the coordinates of one end-point of a diameter of the circle $x^2+y^2-8x-4y+15=0$ are $(2, 1),$ the coordinates of its other end are