Proof of equation of complex number equation of parabola

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How does one derive equation of parabola using complex numberS? I'm talking about of stuff like the representation of circle using complex numbers $|z-z_o|=r$

Edit: I figured out distance from point on parabola to focus is |z-z_o| but how does one write distance from parabola to directix

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Idea: use the fact that a parabola is the locus of points whose distance from a fixed point, say $z_0$, is the same as its distance from a fixed line.

The distance to $z_0$ is easy. For the distance between a point and a line, I suggest reading this related post.