4.2.1—Complex loci

|z−a|=r describes a circle, arg(z−a)=θ a ray, and |z−a|=|z−b| the perpendicular bisector of AB. Each condition is a set of allowed points.

Sketch reference points first, then apply equality or inequality signs. For multiple conditions, keep only their intersection and note excluded endpoints or rays.

|z−(1+i)|=2 is a circle centred at (1,1) with radius 2.

The modulus condition gives distance, while an argument condition gives direction; confusing them changes the locus completely.