Loci in the Argand Diagram

This lesson covers loci — sets of points in the Argand diagram that satisfy a given condition involving a complex variable $z$z. Loci problems are a key part of AQA Further Mathematics and combine algebraic and geometric reasoning.

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Locus Type 1: $|z - z_1| = r$∣z−z1​∣=r (Circle)

The set of all points $z$z satisfying:

$|z - z_1| = r$∣z−z1​∣=r

is a circle with centre $z_1$z1​ and radius $r$r.

Reasoning: $|z - z_1|$∣z−z1​∣ is the distance from $z$z to $z_1$z1​, so the locus is all points at distance $r$r from $z_1$z1​.

Worked Example 1: Sketch the locus $|z - 3 - 2i| = 4$∣z−3−2i∣=4.

Rewrite as $|z - (3 + 2i)| = 4$∣z−(3+2i)∣=4.

This is a circle centred at $(3, 2)$(3,2) with radius 4.

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Locus Type 2: $|z - z_1| = |z - z_2|$∣z−z1​∣=∣z−z2​∣ (Perpendicular Bisector)

The set of all points $z$z satisfying:

$|z - z_1| = |z - z_2|$∣z−z1​∣=∣z−z2​∣

is the perpendicular bisector of the line segment joining $z_1$z1​ and $z_2$z2​.

Reasoning: The points equidistant from $z_1$z1​ and $z_2$z2​ form the perpendicular bisector.