The equation of the chord of contact of the circle $x^2 + y^2 + 4x + 6y - 12 = 0$ with respect to the point $(2, 3)$ is:

The equation of the circle whose diameter is the common chord of the circles $x^{2}+y^{2}+2ax+c=0$ and $x^{2}+y^{2}+2by+c=0$ is

If a variable circle $S=0$ touches the line $y=x$ and passes through the point $(0,0)$,then the fixed point that lies on the common chord of the circles $x^2+y^2+6x+8y-7=0$ and $S=0$ is

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:

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 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