If the lengths of the chords intercepted by the circle ${x^2} + {y^2} + 2gx + 2fy = 0$ from the co-ordinate axes be $10$ and $24$ respectively, then the radius of the circle is..

If the length of the tangents drawn from the point $(1,2)$ to the circles ${x^2} + {y^2} + x + y - 4 = 0$ and $3{x^2} + 3{y^2} - x - y + k = 0$ be in the ratio $4 : 3$, then $k =$

If $2x - 4y = 9$ and $6x - 12y + 7 = 0$ are the tangents of same circle, then its radius will be

Equation of the tangent to the circle, at the point $(1 , -1)$ whose centre is the point of intersection of the straight lines $x - y = 1$ and $2x + y= 3$ is

The line $lx + my + n = 0$ will be a tangent to the circle ${x^2} + {y^2} = {a^2}$ if

The equation to the tangents to the circle ${x^2} + {y^2} = 4$, which are parallel to $x + 2y + 3 = 0$, are