Standard loci in the complex plane

The difference $z-a$ is the displacement from the fixed point a to the moving point z. Therefore $|z-a|$ is their distance and $\arg(z-a)$ is the direction from a to z.

• $|z-a|=r$: circle with centre a and radius r.
• $|z-a|=|z-b|$: perpendicular bisector of the segment joining a and b.
• $\arg(z-a)=\alpha$: ray from a at direction α.

• $|z-a|<r$ is the inside of the circle without the circumference.
• $|z-a|\le r$ includes the circumference.
• $|z-a|<|z-b|$ is the half-plane closer to a than b.
• $\operatorname{Re}z\le k$ is on or left of $x=k$; $\operatorname{Im}z\ge k$ is on or above $y=k$.
• An argument inequality creates a wedge. Test a simple point if the direction of shading is uncertain.

Draw each condition separately, then keep only the points satisfying all conditions. Mark included boundaries solid and excluded boundaries dashed or open.

A ray $\arg(z-a)=\alpha$ starts at a and points in the direction α. The point a itself is excluded because there $z-a=0$, whose argument is undefined.

(Pure Mathematics 2 & 3 Coursebook, Ch. 11.5)