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
(a) On a single Argand diagram sketch the loci given by the equations $|z - 5 - 2\mathrm{i}| = 2$ and $\arg(w + 1) = \tfrac{1}{3}\pi$ , where $z$ and $w$ are complex numbers.

(4 marks)
(b) Hence find the least value of $|z - w|$ for points on these loci. Give your answer in an exact form.

(2 marks)
Level 4 · 6 marksTry it
2
Describe the locus $\arg(z - 2) = \dfrac{\pi}{4}$ , and give the equation of the line on which it lies together with any restriction. (3 marks)
Level 4 · 3 marksTry it
3
Find the Cartesian equation of the locus $|z - 6i| = |z - 2|$ . (4 marks)
Level 4 · 4 marksTry it
4
The complex number $z$ satisfies $\arg(z + 1 - i) = -\dfrac{\pi}{4}$ .
(a) Describe the locus of $z$ . (2 marks)
(b) Find the Cartesian equation of the locus, stating the restriction on $x$ . (3 marks)
Level 4 · 5 marksTry it
5
The complex number $z$ satisfies both

$|z - 4| = |z| \qquad \text{and} \qquad |z - 2| = 3$

Find the two possible values of $z$ . (5 marks)
Level 4 · 5 marksTry it

1 more question on loci in the argand diagram

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