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

In the $xy$-plane,the segment with endpoints $(3, 8)$ and $(-5, 2)$ is the diameter of a circle. The point $(k, 10)$ lies on the circle for:

The equation of the concentric circle,with the circle $C_1$ having equation $x^2+y^2-6x-4y-12=0$ and having double the area compared to the area of $C_1$,is

In $\triangle ABC$,$\angle A = 90^{\circ}$ and the coordinates of points $B$ and $C$ are $(2, -4)$ and $(1, 5)$. Then the equation of the circumcircle of $\triangle ABC$ 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 :

If the lines $x + 2y - 5 = 0$ and $3x - y - 1 = 0$ denote two diameters of a circle of radius $5 \text{ units}$,then the equation of the circle is