The length of the common chord of the circles $x^2 + y^2 + 2x + 3y + 1 = 0$ and $x^2 + y^2 + 4x + 3y + 2 = 0$ is

The chord of contact of the point $(3, 2)$ with respect to the circle $x^2 + y^2 = 25$ meets the coordinate axes at $A$ and $B$. The circumcentre of triangle $OAB$ is

Two circles centered at $(2,3)$ and $(5,6)$ intersect each other. If the radii are equal,the equation of the common chord is

Two circles which touch both the coordinate axes intersect at the points $A$ and $B$. If $A=(1,2)$,then $AB=$

If the circles $x^2+y^2+5kx+2y+k=0$ and $2x^2+2y^2+2kx+3y-1=0$,$k \in R$ intersect at points $P$ and $Q$,then the line $4x+5y-k=0$ passes through $P$ and $Q$ for

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