The Complex Plane

• |z − a| = r: circle, centre a, radius r.
• |z − a| = |z − b|: perpendicular bisector of the line joining a and b.
• |z − a| = k|z − b| (k ≠ 1): a circle — substitute z = x + iy to find its centre and radius.
• arg(z − a) = β: half-line from a (a itself excluded); arg((z − a)/(z − b)) = β: an arc of a circle through a and b.
• Transformations w = f(z): make z the subject, then substitute into the given z-locus.

Distance and direction

For a locus like |z − a| = k|z − b|, put z = x + iy, square both moduli, and complete the square to find the circle.

Shading inequalities

Replace "=" by "<" or "≤" and shade the side that satisfies the inequality: |z − a| ≤ r is the inside of the circle; |z − a| ≤ |z − b| is the half-plane on a's side of the perpendicular bisector; α ≤ arg(z − a) ≤ β is a sector. For a combination, shade the overlap and test one point.

Mapping a locus from z to w

Given w = f(z) and a locus in the z-plane: rearrange to get z in terms of w, substitute into the z-locus, and simplify into a locus in w (often by putting w = u + iv).