Two circles of equal radius $a$ cut orthogonally. If their centres are $(2, 3)$ and $(5, 6)$,then the radical axis of these circles passes through the point:

The number of common tangents that can be drawn to the circles $x^2+y^2-2x-2y-23=0$ and $x^2+y^2-4x-4y-1=0$ is

$A$ circle $S$ passes through the points of intersection of the circles $x^2+y^2-2x+2y-2=0$ and $x^2+y^2+2x-2y+1=0$. If the centre of this circle $S$ lies on the line $x-y+6=0$,then the radius of the circle $S$ is

The line $x+y+2=0$ intersects the circle $x^2+y^2+4x-4y-4=0$ at two points $A$ and $B$. Let $S \equiv x^2+y^2+2gx+2fy+c=0$ be a different circle passing through the points $A$ and $B$. If the distance of the centre of $S=0$ from $AB$ is $\sqrt{2}$,then $g+f+c=$

If the circles $x^{2}+y^{2}=9$ and $x^{2}+y^{2}+2 \alpha x+2 y+1=0$ touch each other internally,then $\alpha$ is equal to

If the points of intersection of the ellipses $x^{2}+2y^{2}-6x-12y+23=0$ and $4x^{2}+2y^{2}-20x-12y+35=0$ lie on a circle of radius $r$ and centre $(a, b)$, then the value of $ab+18r^{2}$ is: