I have the complex equation of an ellipse: $$|z-p-i|+|z+p-i|=q$$ where $z$ is an element of $\mathbb{C}$, and where $p$ and $q$ are positive real numbers.
According to the equation, it should equal to $x^2+4(y-1)^2=1$. That is, I'm being asked to find $p$ and $q$.
Thanks.
Given two focal points $e_1$ and $e_2$, an ellipse is given by equation $$d(z,e_1) + d(z,e_2) \equiv \text{const.}$$
Your two focal points should be $e_1 = - p - i$, $e_2 = p - i$ and therefore the focal length is $e = \frac 12|e_1-e_2| = p.$
If you consider the rightmost (or the leftmost) point on an ellipse, the sum of the distances from focal points is precisely $2a$, where $a$ is semi-major axis, so the equation of an ellipse becomes
$$d(z,e_1) + d(z,e_2) = 2a.$$
Therefore, extract $a$ and $e$ from the equation $x^2+4(y-1)^2 = 1$ to get $p$ and $q$.