4.5.4—Complex loci

Write z=x+iy and translate modulus or argument conditions into equations or inequalities in x and y. The resulting curve should match the geometric interpretation.

For |z−a|=r, expand to a circle; for equal distances, subtract squared distances to obtain a line. Check restrictions introduced by squaring or arguments.

|z−(1+i)|=|z−(−1+i)| simplifies to x=0, the vertical bisector of the two centres.

Squaring distances is safe for non-negative moduli, but argument equations still need branch and quadrant checks.