Two concentric circles are such that the smaller divides the larger into two regions of equal area. If the radius of the smaller circle is $2$ , then the length of the tangent from any point $' P '$ on the larger circle to the smaller circle is :

The equations of the normals to the circle ${x^2} + {y^2} - 8x - 2y + 12 = 0$ at the points whose ordinate is $-1,$ will be

Two tangents are drawn from the point $\mathrm{P}(-1,1)$ to the circle $\mathrm{x}^{2}+\mathrm{y}^{2}-2 \mathrm{x}-6 \mathrm{y}+6=0$. If these tangents touch the circle at points $A$ and $B$, and if $D$ is a point on the circle such that length of the segments $A B$ and $A D$ are equal, then the area of the triangle $A B D$ is eqaul to:

The point at which the normal to the circle ${x^2} + {y^2} + 4x + 6y - 39 = 0$ at the point $(2, 3)$ will meet the circle again, is

If the point $(1, 4)$ lies inside the circle $x^2 + y^2-6x - 10y + p = 0$ and the circle does not touch or intersect the coordinate axes, then the set of all possible values of $p$ is the interval

The line $x = y$ touches a circle at the point $(1, 1)$. If the circle also passes through the point $(1, -3)$, then its radius is