If the length of the tangent from $(f, g)$ to the circle $x^2+y^2=6$ is twice the length of the tangent from the same point to the circle $x^2+y^2+3x+3y=0$,then $f^2+g^2+4f+4g+2$ is equal to

The power of a point $(2, -1)$ with respect to a circle $C$ of radius $4$ is $9$. The centre of the circle $C$ lies on the line $x+y=0$ and in the $2^{\text{nd}}$ quadrant. If $(\alpha, \beta)$ is the centre of the circle $C$,then $\beta-\alpha=$

The power of a point $(2,0)$ with respect to a circle $S$ is $-4$ and the length of the tangent drawn from the point $(1,1)$ to $S$ is $2$. If the circle $S$ passes through the point $(-1,-1)$,then the radius of the circle $S$ is

If a circle of radius $r$ touches the positive coordinate axes and also the circle $x^2+y^2-12x-10y+52=0$ externally,then the distance between the centres of the two circles is

Let $A(2, 3), B(4, 5)$ be two points and let $C = (x, y)$ be a point such that $(x - 2)(x - 4) + (y - 3)(y - 5) = 0$. If the area of $\Delta ABC = \sqrt{2}$ sq. unit,then the maximum number of positions of $C$ in the $xy$ plane is

If $P(2,3)$ and $Q(-1,2)$ are conjugate points with respect to the circle $x^2+y^2+2gx+3y-2=0$,then the radius of the circle is