Find the equations of the two lines which pass through the point $(0,4)$ and form tangents to a circle of radius $2$, centered on the origin.
Firstly, we have the equation of the circle $x^{2}+y^{2}=4$. Which we can rearrange to get $y$ in terms of $x$:
$y^{2}=4-x^{2}\implies y=\pm\sqrt{4-x^{2}}$
However, we know that the tangents must touch points on the top half of the circle, therefore we can simply take the principle square root, $y=\sqrt{4-x^{2}}$. Moreover, since the lines originate from the same point, and are tangential to the same circle, the two tangential points are $(x,\sqrt{4-x^{2}})$ and $(-x,\sqrt{4-x^{2}})$.
The gradients of the tangents at these points can be found by implictly differentiating the original equation and obtaining:
$\frac{dy}{dx}=\frac{-x}{y}=\left\{\frac{-x}{\sqrt{4-x^{2}}},\frac{x}{\sqrt{4-x^{2}}}\right\}$
However, I'm unsure how to go about completing this problem. I know it's a simple question, but I simply cannot see how to solve it.
Thanks in advance!