Locus in the complex plane given an equation

Question

I have the question

Let $a$ and $e$ be two positive real numbers, with $0 < e < 1$. Describe the locus of the points $z$ in the complex plane which satisfy $|z - ae| + |z + ae| = 2a$.

I know this equation is telling me

(distance from z to (ae, 0)) + (distance from (x, y) to ($-ae$, 0) = 2a

and I know that $0 < e < a$ and $ae < a$.

Inspecting a graph thus reveals two solutions: $z = a$ and $z = -a$. But I haven't been able to make more headway than this.

Will I have to resort to algebra (which looks very messy indeed), or is there a more elegant geometric solution that you can point out?

Answer 1

distance from z to $(ae,0)$ + distance from $(x,y)$ to $(−ae,0)=2a$

...and the geometric shape defined by the property that the sum of the two distances from each of its points to two fixed points (called foci) is constant is called by what name exactly... ? :-)