The length of the tangent from the point $(5, 1)$ to the circle $x^2 + y^2 + 6x - 4y - 3 = 0$ is:

The area of the quadrilateral formed by the tangents from the point $(4,5)$ to the circle $x^2+y^2-4x-2y-11=0$,and the pair of radii joining the points of contact of these tangents is

Let a circle $C$ touch the lines $L_{1}: 4x - 3y + K_{1} = 0$ and $L_{2}: 4x - 3y + K_{2} = 0$,where $K_{1}, K_{2} \in R$. If a line passing through the centre of the circle $C$ intersects $L_{1}$ at $(-1, 2)$ and $L_{2}$ at $(3, -6)$,then the equation of the circle $C$ is:

Statement $1$: The only circle having radius $\sqrt{10}$ and a diameter along the line $2x + y = 5$ is $x^2 + y^2 - 6x + 2y = 0$.
Statement $2$: $2x + y = 5$ is a normal to the circle $x^2 + y^2 - 6x + 2y = 0$.

The equation of a circle touching the curve $|x - 1| + |y - 4| = 6$ is

If two circles $(x - 1)^2 + (y - 3)^2 = r^2$ and $x^2 + y^2 - 8x + 2y + 8 = 0$ intersect in two distinct points,then