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 point of intersection of the direct common tangents drawn to the circles $(x+11)^2+(y-2)^2=225$ and $(x-11)^2+(y+2)^2=25$ is

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

$A$ line $lx + my + n = 0$ meets the circle $x^2 + y^2 = a^2$ at the points $P$ and $Q$. The tangents drawn at the points $P$ and $Q$ meet at $R$. Then,the coordinates of $R$ are:

If the chord $L \equiv y-mx-1=0$ of the circle $S \equiv x^2+y^2-1=0$ touches the circle $S_1 \equiv x^2+y^2-4x+1=0$,then the possible points for which $L=0$ is a chord of contact of $S=0$ are

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 $5a + b =$