The value of $c$, for which the line $y = 2x + c$ is a tangent to the circle ${x^2} + {y^2} = 16$, is

Let the tangents at the points $A (4,-11)$ and $B (8,-5)$ on the circle $x^2+y^2-3 x+10 y-15=0$, intersect at the point $C$. Then the radius of the circle, whose centre is $C$ and the line joining $A$ and $B$ is its tangent, is equal to

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

The angle between the pair of tangents from the point $(1, 1/2)$ to the circle $x^2 + y^2 + 4x + 2y -4=0$ is-

If $\theta $ is the angle subtended at $P({x_1},{y_1})$ by the circle $S \equiv {x^2} + {y^2} + 2gx + 2fy + c = 0$, then

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