|z-1| = |z +i| On Which Sets Of Points For z In The Complex Plane?

Question

On Which Sets Of Points For z In The Complex Plane?

Answer 1

Take $ z= x + iy $ an you will get $(x-1)^2 + y^2 = x^2 + (y+1)^2 $. Now try to simplify this. (after simplifying: $ x+y = 0$)

Answer 2

Hint: the equation is that of the set of points that are equidistant from $z=1$ and $z=-i$