Why is this an ellipse?

Question

On a textbook, I've arrived at the following function:

$\displaystyle \phi(z)=\log{\frac{|z-\sqrt{(z²-1})|}{2}}$

and it says that the formula has a simple interpretation: the level curves of $\phi(z)$ are the ellipses with foci $-1, 1$. I know the problem is reduced to proving $|z-\sqrt{(z²-1})| = k$ is an ellipse, $k$ constant, but I don't know why this is true. So, my question is: Why is this an ellipse?

Answer 1

Without caring about choosing a branch of the square root etc., when $z$ traverses a level curve of $\phi$, then

$$w = z-\sqrt{z^2-1}$$

traverses a circle. So let's identify the inverse mapping of $z \mapsto z - \sqrt{z^2-1}$:

$$\begin{align} w = z - \sqrt{z^2-1} &\iff \sqrt{z^2-1} = z-w\\ &\Rightarrow z^2-1 = z^2-2zw+w^2\\ &\iff 2zw = w^2+1\\ &\iff z = \frac12\left(w+\frac1w\right). \end{align}$$

Now write $w = r e^{i\varphi}$ to see that that maps circles to ellipses.