Two orthogonal circles are such that the area of one is twice the area of the other. If the radius of the smaller circle is $r$,then the distance between their centers will be -

If $A$ and $B$ are the centres of similitude with respect to the circles $x^2+y^2-14x+6y+33=0$ and $x^2+y^2+30x-2y+1=0$,then the midpoint of $AB$ is

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 $3(\alpha + \beta + \gamma)$.

If the circle $x^2+y^2+8x-4y+c=0$ touches the circle $x^2+y^2+2x+4y-11=0$ externally and cuts the circle $x^2+y^2-6x+8y+k=0$ orthogonally,then $k$ is equal to

If the circles $x^2+y^2-6x-8y-12=0$ and $x^2+y^2-4x+6y+k=0$ are orthogonal to each other,then the value of $k$ is:

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