4.2.3—Complex loci

A condition such as |z−a|=|z−b| gives a perpendicular bisector, while |z−a|=r gives a circle and arg(z−a)=θ gives a ray. Multiple conditions mean an intersection of sets.

Translate each condition separately, sketch boundaries, and test a point to select the correct side for inequalities.

|z−1|=|z+1| is the imaginary axis: points equidistant from 1 and −1 have real part zero.

The equal-distance locus is not the line joining the two points; it is perpendicular to that line.