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

Loci and regions on the Argand plane describe the geometric paths and areas traced by a variable complex number z = x + iy satisfying specified equations or inequalities: (1) Circle: |z - z₀| = r represents a circle with centre z₀ and radius r; |z - z₀| ≤ r represents the closed circular disc; (2) Perpendicular Bisector: |z - z₁| = |z - z₂| represents the straight line perpendicularly bisecting the segment connecting z₁ and z₂; |z - z₁| < |z - z₂| denotes the half-plane closer to z₁; (3) Half-Line / Ray: Arg(z - z₀) = α represents a ray originating at z₀ (excluding z₀) inclined at angle α to the positive real axis; (4) Sector / Annulus: Combined bounds such as r₁ ≤ |z - z₀| ≤ r₂ and α ≤ Arg(z - z₀) ≤ β represent annular sectors.

Check before you answer (A/L Pure Maths Paper 1)

1. Boundary Inclusions: Using solid lines for non-strict inequalities (≤, ≥) and dashed lines for strict inequalities (<, >) with clear exclusion of origin points on rays.
2. Cartesian Verification: Verifying geometric interpretations by substituting z = x + iy and deriving the equivalent Cartesian Cartesian line or circle equation.
3. Shading Precision: Correctly identifying and shading the test point (e.g., origin z = 0) to confirm the correct region satisfying multi-constraint inequalities.