The number of tangents which can be drawn from the point $(-1,2)$ to the circle ${x^2} + {y^2} + 2x - 4y + 4 = 0$ is

$x = 7$ touches the circle ${x^2} + {y^2} - 4x - 6y - 12 = 0$, then the coordinates of the point of contact are

Consider a circle $(x-\alpha)^2+(y-\beta)^2=50$, where $\alpha, \beta>0$. If the circle touches the line $y+x=0$ at the point $P$, whose distance from the origin is $4 \sqrt{2}$ , then $(\alpha+\beta)^2$ is equal to................

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

Suppose two perpendicular tangents can be drawn from the origin to the circle $x^2+y^2-6 x-2 p y+17=0$, for some real $p$. Then, $|p|$ is equal to

Which of the following lines is a tangent to the circle ${x^2} + {y^2} = 25$ for all values of $m$.....