Tangents $AB$ and $AC$ are drawn from the point $A(0, 1)$ to the circle $x^2 + y^2 - 2x + 4y + 1 = 0$. The equation of the circle passing through $A, B,$ and $C$ is

The radius of the circle,having centre at $(2, 1)$ and one of its chords as a diameter of the circle $x^2 + y^2 - 2x - 6y + 6 = 0$,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 $PQ=$

The length of the chord joining the points in which the straight line $\frac{x}{3} + \frac{y}{4} = 1$ cuts the circle ${x^2} + {y^2} = \frac{169}{25}$ is

Tangents drawn from the origin $O$ to the circle $x^2 + y^2 + 2gx + 2fy + c = 0$ touch the circle at the points $P$ and $Q$. Then the equation of the circumcircle of the triangle $OPQ$ is

If the circles $x^{2}+y^{2}-2x-2y-7=0$ and $x^{2}+y^{2}+4x+2y+k=0$ cut orthogonally,then the length of the common chord of the circles is