Find the locus of points on a complex plane that satisfy |z-3| - |z+3| = 4

Question

Can you find the locus of points on the complex plane represented by the equation below:

|z-3| - |z+3| = 4

Answer 1

Hint:

Given two points called the foci (plural of focus), which are $(c,0)$ and $(-c,0)$ in the picture below, a hyperbola is precisely the locus of points $(x,y)$ such that the difference between $(x,y)$ and the foci is constant.

Since the complex plane can be visualized like an $xy$-plane, this should help. Say $z=x+iy$. Then your equation says that the distance between $(x,y)$ and $(3,0)$ minus the distance between $(x,y)$ and $(-3,0)$ is constant, equal to 4. This is exactly half of the hyperbola you see below with $c=3$. The equation of this hyperbola is $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, where $a$ is the real number that's the solution of the equation that you should try to find, and $a^2+b^2=c^2$.

Reference and more info: https://courses.lumenlearning.com/waymakercollegealgebra/chapter/equations-of-hyperbolas/#:~:text=Reviewing%20the%20standard%20forms%20given,%3D%20a%202%20%2B%20b%202%20.