The length of the common chord of the circles $x^2 + y^2 - 6x + 5 = 0$ and $x^2 + y^2 - 2y - 3 = 0$ is

If the circles $x^2 + y^2 + 2ax + cy + a = 0$ and $x^2 + y^2 - 3ax + dy - 1 = 0$ intersect at two distinct points $P$ and $Q$,then for what value of $a$ does the line $5x + 6y - a = 0$ pass through points $P$ and $Q$?

The distance between the chords of contact of tangents to the circle $x^2 + y^2 + 2gx + 2fy + c = 0$ from the origin and the point $(g, f)$ is:

If the line $x - 2y = k$ cuts off a chord of length $2$ from the circle ${x^2} + {y^2} = 3$,then $k =$

$A$ rhombus is inscribed in the region common to the two circles $x^2 + y^2 - 4x - 12 = 0$ and $x^2 + y^2 + 4x - 12 = 0$,with two of its vertices on the line joining the centers of the circles. The area of the rhombus is

If the two circles $x^2+y^2-2x-6y+10-r^2=0$ and $x^2+y^2-8x+2y+8=0$ have a common chord of non-zero length,then