Find the centre and radius of the circle $x^{2}+y^{2}-8x+10y-12=0$.

The radius of the larger circle lying in the first quadrant and touching the line $4x + 3y - 12 = 0$ and the coordinate axes is:

Find the equation of a circle whose center lies on the $y$-axis,has a radius of $3$,and passes through the origin.

The polar equation of the circle with centre $\left(2, \frac{\pi}{2}\right)$ and radius $3$ units is :

The lines $2x - 3y = 5$ and $3x - 4y = 7$ are the diameters of a circle of area $154$ square units. The equation of the circle is

Let $S$ be a circle concentric with the circle $3x^2+3y^2+x+y-1=0$. If the length of the tangent drawn from a point $(2,-2)$ to the given circle is the radius of the circle $S$,then the power of the point $(2,1)$ with respect to the circle $S$ is