Tangents are drawn from the point $(-1,-4)$ to the circle $x^2 + y^2 - 2x + 4y + 1 = 0$. Length of corresponding chord of contact will be-
$\sqrt 2$ units
$2\sqrt 2$ units
$3\sqrt 2$ units
$2$ units
If the distances from the origin to the centres of the three circles ${x^2} + {y^2} - 2{\lambda _i}\,x = {c^2},(i = 1,\,2,\,3)$ are in $G. P.$, then the lengths of the tangents drawn to them from any point on the circle ${x^2} + {y^2} = {c^2}$ are in
The condition that the line $x\cos \alpha + y\sin \alpha = p$ may touch the circle ${x^2} + {y^2} = {a^2}$ is
The length of the tangent from the point $(4, 5)$ to the circle ${x^2} + {y^2} + 2x - 6y = 6$ is
The equations of the tangents drawn from the point $(0, 1)$ to the circle ${x^2} + {y^2} - 2x + 4y = 0$ are
If the line $y = \sqrt 3 x + k$ touches the circle ${x^2} + {y^2} = 16$, then $k =$