Two circles centered at $(2,3)$ and $(4,5)$ intersect each other. If their radii are equal,then the equation of the common chord is

Find the distance between the chords of contact of the circle $x^2 + y^2 + 2gx + 2fy + c = 0$ with respect to the points $(0, 0)$ and $(g, f)$.

$y = mx$ is a chord of a circle of radius $a$. The diameter of the circle lies along the $x$-axis and one end of this chord is at the origin. The equation of the circle described on this chord as diameter is

The length of the common chord of the circles $x^2+y^2-6x-4y+9=0$ and $x^2+y^2-8x-6y+23=0$ is

Let $P$ and $Q$ be two external points of the circle $S \equiv x^2+y^2-a^2=0$. Let the chord of contact of the point $P$ with respect to the circle $S=0$ pass through $Q$. If $l_1$ and $l_2$ are the lengths of the tangents drawn from $P$ and $Q$ to the circle $S=0$,then $PQ=$

If the circle $x^2 + y^2 = a^2$ cuts off a chord of length $2b$ from the line $y = mx + c$,then