The equation of the tangents to the circle ${x^2} + {y^2} + 4x - 4y + 4 = 0$ which make equal intercepts on the positive coordinate axes is given by

The equation of tangent to the circle ${x^2} + {y^2} = {a^2}$ parallel to $y = mx + c$ is

Let the lengths of intercepts on $x$ -axis and $y$ -axis made by the circle $x^{2}+y^{2}+a x+2 a y+c=0$ $(a < 0)$ be $2 \sqrt{2}$ and $2 \sqrt{5}$, respectively. Then the shortest distance from origin to a tangent to this circle which is perpendicular to the line $x +2 y =0,$ is euqal to :

The equation of the circle whose radius is $5$ and which touches the circle ${x^2} + {y^2} - 2x - 4y - 20 = 0$ externally at the point $(5, 5)$ is

The equation of pair of tangents to the circle ${x^2} + {y^2} - 2x + 4y + 3 = 0$ from $(6, - 5)$, is

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 $k$ =