A-level · Complex numbers

Loci in the Argand diagramExam-style questions

Reading a condition on z as a set of points in the Argand diagram: |z − a| = r is a circle centre a radius r, |z − a| = |z − b| is the perpendicular bisector of a and b, and arg(z − a) = θ is a half-line from a. Describing and sketching these loci, turning them into Cartesian equations, and shading the regions given by the matching inequalities.

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16 questions · A Level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 1 Level 2 Level 3 Level 4 Level 5

1
The locus of $z$ satisfies $|z + 2| = |z - 4|$ .
(a) Describe the locus geometrically. (1 mark)
(b) Find its Cartesian equation. (3 marks)
Level 3 · 4 marksTry it
2
Show that the locus $|z - 1| = |z + i|$ has Cartesian equation $y = -x$ . (3 marks)
Level 3 · 3 marksTry it
3
The complex number $z$ satisfies $|z - 3 - 4i| \leqslant 2$ .
(a) Describe the region represented by this inequality. (2 marks)
(b) Find the least and greatest values of $|z|$ . (2 marks)
Level 3 · 4 marksTry it
4
Find the Cartesian equation of the locus $|z - 2i| = |z - 4|$ , giving your answer in the form $y = mx + c$ . (4 marks)
Level 3 · 4 marksTry it
5
(a) Describe the locus $\mathrm{Re}(z) = 3$ . (1 mark)
(b) Find the Cartesian equation of the locus $\mathrm{Im}\big((1 + i)z\big) = 2$ . (3 marks)
Level 3 · 4 marksTry it