Then take the absolute value of x + iy - 3i = x + i(y - 3) and x + iy + 3i = x + i(y + 3).

So you have to simplify something of the form .

Square both sides, simplify, square both sides again and simplify. You will get asinglebranch of a hyperbola (there is an implicit restriction on z you will need to be careful of that. This restriction means that both branches do no get included).

Of course, if you're familiar with the locus definition of a hyperbola then there's a much simpler geometric approach that can be taken.