CAIE A-Level Mathematics 3.9.8 Complex-Number Loci

CAIE A-Level Mathematics 3.9.8 Complex-Number Loci
Cambridge International AS & A Level Mathematics 9709 syllabus for exams in 2028, 2029 and 20302028–2030

Practise converting complex equations and inequalities into lines, circles, half-planes and angle regions on Argand diagrams and finding extreme modulus or argument values.

How this is tested

  • translate Re z, Im z, |z − a| and arg(z − a) into the corresponding geometric boundary
  • use strict or inclusive inequalities to select the correct side, disc or angular region
  • locate tangent or boundary points to calculate maximum or minimum |z| or arg z

Question 4

[Maximum number: 5]

On an Argand diagram shade the region whose points represent complex numbers z which satisfy both the inequalities ∣z+2i∣⩽3|z+2 \mathrm{i}| \leqslant 3 and ∣z+2i∣⩽∣z−2+4i∣|z+2 \mathrm{i}| \leqslant|z-2+4 \mathrm{i}|.