$A$ circle $C$ touches the line $x=2y$ at the point $(2,1)$ and intersects the circle $C_{1}: x^{2}+y^{2}+2y-5=0$ at two points $P$ and $Q$ such that $PQ$ is a diameter of $C_{1}$. Then the diameter of $C$ is:

If the angle between the circles $x^2+y^2-2x+ky+1=0$ and $x^2+y^2-kx-2y+1=0$ is $\cos^{-1}(\frac{1}{4})$ and $k < 0$,then the point which lies on the radical axis of the given circles is

If the radical centre of the three circles $x^2+y^2=1$,$x^2+y^2-2x-3=0$,and $x^2+y^2-2y-3=0$ is $C(\alpha, \beta)$ and $r$ is the sum of the radii of the given circles,then the equation of the circle with $C(\alpha, \beta)$ as centre and $r$ as radius is:

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

The tangent to the circle $x^2 + y^2 = 5$ at the point $(1, -2)$ intersects the circle $x^2 + y^2 - 8x + 6y + 20 = 0$ at which of the following?

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