Which of the following is a point on the common chord of the circles ${x^2 + y^2 + 2x - 3y + 6 = 0}$ and ${x^2 + y^2 + x - 8y - 13 = 0}$?

The locus of the middle points of those chords of the circle $x^2 + y^2 = 4$ which subtend a right angle at the origin is

Let the circle $C_1: x^2+y^2-2(x+y)+1=0$ and $C_2$ be a circle having centre at $(-1,0)$ and radius $2$. If the line of the common chord of $C_1$ and $C_2$ intersects the $y$-axis at the point $P$,then the square of the distance of $P$ from the centre of $C_1$ is:

The length of the common chord of the circles $x^2+y^2+3x+5y+4=0$ and $x^2+y^2+5x+3y+4=0$ is

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

Let $P$ be any point on the circle $x^2+y^2-2x-1=0$ and $C$ be its centre. Let $AB$ be the chord of contact of $P$ with respect to the circle $x^2+y^2-2x=0$. Then the locus of the circumcentre of the triangle $CAB$ is