A circle passes through the points $(- 1, 1) , (0, 6)$ and $(5, 5)$ . The point$(s)$ on this circle, the tangent$(s)$ at which is/are parallel to the straight line joining the origin to its centre is/are :

If $OA$ and $OB$ be the tangents to the circle ${x^2} + {y^2} - 6x - 8y + 21 = 0$ drawn from the origin $O$, then $AB =$

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

If variable point $(x, y)$ satisfies the equation $x^2 + y^2 -8x -6y + 9 = 0$ , then range of $\frac{y}{x}$ is

The tangent at $P$, any point on the circle ${x^2} + {y^2} = 4$, meets the coordinate axes in $A$ and $B$, then

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