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 line joining the centres of the circles belonging to the coaxial system of circles $4x^2 + 4y^2 - 12x + 6y - 3 + \lambda(x + 2y - 6) = 0$ is

If the line $y = x + 3$ intersects the circle $x^2 + y^2 = a^2$ at two points $A$ and $B$,then the equation of the circle having $AB$ as its diameter is . . . . . .

Find the equation of a circle with radius $5$ units and touching the circle $x^2+y^2-2x-4y-20=0$ at the point $(5,5)$.

If the cosine of the angle between the two circles $x^2+y^2+2x+4y-3=0$ and $x^2+y^2+2kx-2y-1=0$ is $\frac{1}{2\sqrt{3}}$,then $k^2=$

Consider the three circles: $S_{1} \equiv x^{2}+y^{2}-6x-6y+4=0$,$S_{2} \equiv x^{2}+y^{2}-2x-4y+3=0$,and $S_{3} \equiv x^{2}+y^{2}+2kx+2y+1=0$. If the radical centre of these three circles exists,then which of the following cannot be the value of $k$?