Loci and regions of a complex number — A/L Combined Mathematics

What must I remember about Loci and regions of a complex number?

• Locus of a variable complex number z = x + iy: The path or region traced out in the Argand plane by z satisfying a given algebraic or geometric condition.
• Circle locus: |z - z0| = r represents a circle with center z0 and radius r; |z - z0| ≤ r represents the closed circular disc including boundary.
• Perpendicular bisector locus: |z - z1| = |z - z2| represents the perpendicular bisector of the line segment joining points z1 and z2.
• Ray / Half-line locus: Arg(z - z0) = α represents a ray (half-line) emanating from z0 making angle α with the positive real axis (point z0 itself excluded).
• Apollonius Circle: |z - z1| / |z - z2| = k (k > 0, k ≠ 1) represents an Apollonius Circle; if k = 1, it degenerates into the perpendicular bisector.

Test yourself (1 minute)

1. Circular arc locus

   Arg[(z - z1)/(z - z2)] = α represents an arc of a circle passing through z1 and z2 subtending angle α; if α = ±π/2, the locus is a semicircle with diameter z1z2; if α = 0 or π, it represents the straight line through z1, z2 (excluding segment).

2. Ellipse and Hyperbola loci

   |z - z1| + |z - z2| = 2a (with 2a > |z1 - z2|) represents an Ellipse with foci z1, z2; ||z - z1| - |z - z2|| = 2a (with 2a < |z1 - z2|) represents a Hyperbola.

3. Open vs closed boundary convention

   When sketching inequalities such as |z - z0| < r or Arg(z) < α, use dashed lines for strict inequalities (<, >) and solid lines for non-strict inequalities (≤, ≥).

Practise Loci and regions of a complex number questions today

Source: NIE G.C.E. (A/L) Combined Mathematics Resource Book - Pure Mathematics, Unit 21

What should I study first?

What comes next?

Next in the syllabus

Where does Loci and regions of a complex number fit in the syllabus?

Year 2 · Term 1