If $(a, b)$ is the common point for the circles $x^2+y^2-4x+4y-1=0$ and $x^2+y^2+2x-4y+1=0$,then $a^2+b^2=$

The equation of the circle passing through $(1,2)$ and the points of intersection of the circles $x^2+y^2-8x-6y+21=0$ and $x^2+y^2-2x-15=0$ is:

The radical axis of the co-axial system of circles with limiting points $(1, 2)$ and $(-2, 1)$ is

The coordinates of the centre of the circle which bisects the circumferences of the circles $x^2 + y^2 = 1$,$x^2 + y^2 + 2x - 3 = 0$,and $x^2 + y^2 + 2y - 3 = 0$ are

Consider the system of circles $x^2+y^2+2fy+\lambda(x^2+y^2+2gx+k)=0$,where $g \neq 0, f \neq 0$ and $\lambda$ is a parameter. If $A$ and $B$ are the point circles of this system such that $\angle AOB = \frac{\pi}{2}$,then $g^2$ is equal to

$A$ circle $S \equiv x^2+y^2-16=0$ intersects another circle $S^{\prime}=0$ of radius $5$ units such that their common chord is of maximum length. If the slope of that chord is $\frac{3}{4}$,then the centre of such a circle $S^{\prime}=0$ is