What is $z_o$ in the equation of parabola in complex plane?

Question

In the equation of parabola in complex plane i.e. $|z-z_0|=\Re{(z)}$ what does $z_o$ mean ?

Answer 1

Let $z = x + iy$. Then the equation $|z - z_0| = \Re(z)$, which has solutions for $\Re(z) \geq 0$, is equivalent to

$$ y = \Im(z_0) \pm \sqrt{2\Re(z_0)(x - \Re(z_0)/2)} $$

You will observe that $z_0$ serves to shift the parabola vertically by its imaginary part, shifts it horizontally by half its real part, and also scales it by its real part. If $\Re(z_0) = 0$, then the parabola reduces to line parallel to the real axis, originating at $(0,\Im(z_0))$ and extending into the the right half plane. As $\Re(z_0)$ increases from zero, the parabola shifts to the right and "fans out". Since the expression under the radical sign must be nonnegative, it is necessary that $\Re(z_0) \geq 0$.