For what value of $k$ do the circles $x^2 + y^2 + 5x + 3y + 7 = 0$ and $x^2 + y^2 - 8x + 6y + k = 0$ intersect orthogonally?

Find the value of $m+n$,if the circumference of the circle $x^2+y^2+8x+8y-m=0$ is bisected by the circle $x^2+y^2-2x+4y+n=0$.

If the line $x+y=2$ cuts the circle $x^2+y^2+2x-4y+4=0$ at two points $A$ and $B$,then the radius of the circle passing through $A$ and $B$ and orthogonal to $x^2+y^2-2x-4y-4=0$ is

The equation of the circle which passes through the origin,has its centre on the line $x + y = 4$ and cuts the circle ${x^2} + {y^2} - 4x + 2y + 4 = 0$ orthogonally,is

Find the radical center of the three circles $x^2 + y^2 = a^2$,$(x - c)^2 + y^2 = a^2$,and $x^2 + (y - b)^2 = a^2$.

Suppose the circle $S: x^2+y^2+2gx+2fy+c=0$ cuts orthogonally the two circles $S': x^2+y^2-4x-6y+11=0$ and $S'': x^2+y^2-10x-4y+21=0$. If the centre of $S=0$ lies on the bisector of the angle between the positive coordinate axes,then $2g+2f+c=$