If the point of intersection of the tangents drawn at the points where the line $5x + y + 1 = 0$ cuts the circle $x^2 + y^2 - 2x - 6y - 8 = 0$ is $(a, b)$,then $5a + b =$

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

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

The point of intersection of the tangents drawn at the points where the line $2x - y + 3 = 0$ meets the circle $x^2 + y^2 - 4x - 6y + 4 = 0$ is

If the angle of intersection at a point where the two circles with radii $5 \ cm$ and $12 \ cm$ intersect is $90^o$,then the length (in $cm$) of their common chord is

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