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

If the parametric equations of the circle passing through the points $(3,4)$,$(3,2)$,and $(1,4)$ are $x=a+r \cos \theta$ and $y=b+r \sin \theta$,then find the value of $b^{a} r^{a}$.

The equation of the circle passing through the origin,whose center lies in the first quadrant and which makes intercepts of length $6$ and $4$ on the $x$-axis and $y$-axis respectively,is:

Find the equation of the circle whose center is $(3, 5)$ and radius is $4$.

Circles are drawn through the point $(2, 0)$ to cut an intercept of length $5$ units on the $x$-axis. If their centres lie in the first quadrant,then their equation is

Let the circle $S: 36 x^{2}+36 y^{2}-108 x+120 y+C=0$ be such that it neither intersects nor touches the coordinate axes. If the point of intersection of the lines $x-2 y=4$ and $2 x-y=5$ lies inside the circle $S$,then :