Let a circle $C$ touch the lines $L_{1}: 4 x-3 y+K_{1}$ $=0$ and $L _{2}: 4 x -3 y + K _{2}=0, K _{1}, K _{2} \in R$. 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

Let the lengths of intercepts on $x$ -axis and $y$ -axis made by the circle $x^{2}+y^{2}+a x+2 a y+c=0$ $(a < 0)$ be $2 \sqrt{2}$ and $2 \sqrt{5}$, respectively. Then the shortest distance from origin to a tangent to this circle which is perpendicular to the line $x +2 y =0,$ is euqal to :

The area of triangle formed by the tangent, normal drawn at $(1,\sqrt 3 )$ to the circle ${x^2} + {y^2} = 4$ and positive $x$-axis, is

If the line $3x + 4y - 1 = 0$ touches the circle ${(x - 1)^2} + {(y - 2)^2} = {r^2}$, then the value of $r$ will be

The square of the length of the tangent from $(3, -4)$ on the circle ${x^2} + {y^2} - 4x - 6y + 3 = 0$ is

The equations of the tangents to the circle ${x^2} + {y^2} = 50$ at the points where the line $x + 7 = 0$  meets it, are