The angle between the tangents to the circle ${x^2} + {y^2} = 169$ at the points $(5, 12) $ and $(12, -5)$ is ............. $^o$

If a circle of radius $R$ passes through the origin $O$ and intersects the coordinate axes at $A$ and $B,$ then the locus of the foot of perpendicular from $O$ on $AB$ 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 $k =$

The line $x\cos \alpha + y\sin \alpha = p$will be a tangent to the circle ${x^2} + {y^2} - 2ax\cos \alpha - 2ay\sin \alpha = 0$, if $p = $

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

If the ratio of the lengths of tangents drawn from the point $(f,g)$ to the given circle ${x^2} + {y^2} = 6$ and ${x^2} + {y^2} + 3x + 3y = 0$ be $2 : 1$, then