Find the locus of points that satisfy

Question

(a) |z+1| = |z+i|
(b) Re z = |z-1|

and sketch them in the complex plane.

Answer 1

since |z-c|=r is a circle of radius r,

|z+1| = |z+i| is the intersection of two circles.

If z = x+yi,
Re z = x
|z-1|^2 = (x-1)^2 + y^2
So, you need x^2 = (x-1)^2

Answer 2

correction on #1. What I gave was the solution for a particular radius. Since we have to consider all possible such circles, the locus is the perpendicular bisector of the line joining centers of the circles. All points on that line are equidistant from the two centers.