Complex loci

Objective. Interpret distance and angle constraints in the Argand plane.

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Learn: the key idea

A complex locus is the set of points in the Argand plane satisfying a condition on z, often expressed using |z − a| or arg(z − a). The equation |z − a| = r describes a circle centre a with radius r, since it fixes the distance from a fixed point. The equation |z − a| = |z − b| describes the perpendicular bisector of the segment joining a and b, since it fixes equal distances from two points. An equation arg(z − a) = θ describes a half-line from a at a fixed angle. For example, |z − 2i| = 3 is a circle centre (0,2) radius 3.

Worked example

Describe fully the locus of points satisfying |z − 3| = |z + 1| in the Argand plane.

1 $|z-3| is distance from the point (3,0); |z+1| is distance from (-1,0)$
2 $The equation states these two distances are equal$
3 $Such points form the perpendicular bisector of the segment joining (3,0) and (-1,0)$
4 $The midpoint is (1,0), and the segment is horizontal, so the bisector is the vertical line x = 1$

Answer: The locus is the vertical line x = 1

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Common mistake. Students often treat |z − a| = r as a full line or forget the locus is only a half-line (not a full line) when arg(z − a) = θ is given, since θ fixes direction, not its opposite. Always sketch the starting point and direction.

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