How can I solve an equation like this:
I've tried expanding z as
and computing the equation
but this way it quickly gets terribly complicated. Plus I have two variables in one expression.
Any other ideas?
Ah, no, there won't. I've forgotten that the task was "mark in the complex plane all numbers satisfying following condition".
So maybe I don't need to calculate anything and just draw? But what? I have a feeling it will be an ellipse but I've never learnt about it.
If you want to avoid doing the algebra, then you can recognise it's an ellipse by noting that:
1. geometrically, |z + 1| is the distance of z from -1.
2. geometrically, |z - 1| is the distance of z from 1.
3. geometrically, |z + 1| + |z - 1| = 3 means (distance of z from -1) + (distance of z from 1) = 3.
4. The above is a well known locus definition of the ellipse (the foci are at z = 1 and z = -1). The cartesian of the ellipse is easily constructed from this: where and ....