If $OA$ and $OB$ are the tangents to the circle $x^2 + y^2 - 6x - 8y + 21 = 0$ drawn from the origin $O$,then $AB =$

The locus of the point of intersection of the tangents at the extremities of a chord of the circle $x^2 + y^2 = a^2$ which touches the circle $x^2 + y^2 = 2ax$ is

The equation of the circle whose diameter is the common chord of the circles $x^2+y^2-3x+y-10=0$ and $x^2+y^2-x+2y-20=0$ is

The length of the common chord of the circles of radii $15$ and $20$,whose centers are $25$ units of distance apart,is

The equation of the pair of tangents drawn from the point $(6, -5)$ to the circle $x^2 + y^2 - 2x + 4y + 3 = 0$ is:

If the circle $S \equiv x^2+y^2-4=0$ intersects another circle $S^{\prime}=0$ of radius $\frac{5 \sqrt{2}}{2}$ in such a manner that the common chord is of maximum length with slope equal to $\frac{1}{4}$,then the centre of $S^{\prime}=0$ is