The co-axial system of circles given by ${x^2} + {y^2} + 2gx + c = 0$ for $c < 0$ represents

If a diameter of the circle $x^2+y^2-4x+6y-12=0$ is a chord of a circle $S$ whose centre is at $(-3, 2)$,then the radius of $S$ is

Find the equation of a circle which passes through the point $(1,2)$ and the points of intersection of the circles $x^2+y^2-8x-6y+21=0$ and $x^2+y^2-2x-15=0$.

The radical axis of the circles $S_1: x^2+y^2-4x+6y-10=0$ and $S_2: x^2+y^2+2x-6y+2=0$ cuts the circle $S_1$ in

If the equation of the circle passing through the points of intersection of the circles $S_1: x^2 - 2x + y^2 - 4y - 4 = 0$ and $S_2: x^2 + 2x + y^2 + 4y - 4 = 0$ passes through the point $(3, 3)$,and its equation is $x^2 + y^2 + \alpha x + \beta y + \gamma = 0$,then find the value of $3(\alpha + \beta + \gamma)$.

The number of common tangents to the circles ${x^2} + {y^2} - 4x - 6y - 12 = 0$ and ${x^2} + {y^2} + 6x + 18y + 26 = 0$ is