If the straight line $y = mx + c$ touches the circle ${x^2} + {y^2} - 4y = 0$, then the value of $c$ will be
$1 + \sqrt {1 + {m^2}} $
$1 - \sqrt {{m^2} + 1} $
$2(1 + \sqrt {1 + {m^2}} )$
$2 + \sqrt {1 + {m^2}} $
From any point on the circle ${x^2} + {y^2} = {a^2}$ tangents are drawn to the circle ${x^2} + {y^2} = {a^2}{\sin ^2}\alpha $, the angle between them is
A circle $C_{1}$ passes through the origin $O$ and has diameter $4$ on the positive $x$-axis. The line $y =2 x$ gives a chord $OA$ of a circle $C _{1}$. Let $C _{2}$ be the circle with $OA$ as a diameter. If the tangent to $C _{2}$ at the point $A$ meets the $x$-axis at $P$ and $y$-axis at $Q$, then $QA : AP$ is equal to.
The equation to the tangents to the circle ${x^2} + {y^2} = 4$, which are parallel to $x + 2y + 3 = 0$, are
The gradient of the tangent line at the point $(a\cos \alpha ,a\sin \alpha )$ to the circle ${x^2} + {y^2} = {a^2}$, is
The equation of tangent to the circle ${x^2} + {y^2} = {a^2}$ parallel to $y = mx + c$ is