The distance of the origin from the external centre of similitude for the circles $x^2+y^2-8x-10y-8=0$ and $x^2+y^2+2x-2y-2=0$ is

If $P(\alpha, \beta)$ is the radical centre of the circles $S \equiv x^2+y^2+4x+7=0$,$S^{\prime} \equiv 2x^2+2y^2+3x+5y+9=0$ and $S^{\prime \prime} \equiv x^2+y^2+y=0$,then the length of the tangent drawn from $P$ to $S^{\prime}=0$ is

The equation of the circle having the chord $x \cos \alpha + y \sin \alpha = p$ of the circle $x^2 + y^2 = a^2$ as its diameter is:

In the co-axial system of circles $x^2 + y^2 + 2gx + c = 0$,where $g$ is a parameter,if $c > 0$,then the circles are

If the acute angle between the circles $S \equiv x^2+y^2+2kx+4y-3=0$ and $S' \equiv x^2+y^2-4x+2ky+9=0$ is $\cos^{-1}(\frac{3}{8})$ and the centre of $S'=0$ lies in the first quadrant,then the radical axis of $S=0$ and $S'=0$ is

Find the equation of the circle passing through the intersection of the circles $x^{2} + y^{2} - 8x - 2y + 7 = 0$ and $x^{2} + y^{2} - 4x + 10y + 8 = 0$,and having its center on the $y-$axis.