If the length of tangent drawn from the point $(5, 3)$ to the circle ${x^2} + {y^2} + 2x + ky + 17 = 0$ be $7$, then $k$ =

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 $x^2 + y^2 -6x+2y = 0$.

The angle of intersection of the circles ${x^2} + {y^2} - x + y - 8 = 0$ and ${x^2} + {y^2} + 2x + 2y - 11 = 0,$ is

The tangent$(s)$ from the point of intersection of the lines $2x -3y + 1$ = $0$ and $3x -2y -1$ = $0$ to circle $x^2 + y^2 + 2x -4y$ = $0$ will be -

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

Line $y = x + a\sqrt 2 $ is a tangent to the circle ${x^2} + {y^2} = {a^2}$ at