What is the locus |z-a|=|z-b|?

The perpendicular bisector of the line joining a and b.

What is the locus \arg(z-a)=\theta?

A half-line from a (excluding a) at angle \theta to the positive real direction.

What is the locus |z-a|=k|z-b| for k\neq1?

A circle (circle of Apollonius).

How do you find the Cartesian equation of |z-a|=k|z-b|?

Put z=x+iy, square both sides and complete the square.

What is \arg\left(\frac{z-a}{z-b}\right) in terms of arguments?

\arg(z-a)-\arg(z-b), the angle at z subtended by a and b.

Locus of \arg\left(\frac{z-a}{z-b}\right)=\beta?

An arc of a circle through a and b (end points excluded).

Which arc if \beta=\frac{\pi}{2}?

A semicircle with diameter from a to b.

Region p\leqRe(z)\leq q?

A vertical strip between x=p and x=q.

Region \alpha\leq\arg(z-z1)\leq\beta?

The wedge between two half-lines from z1.

Greatest |z| on a circle centre c radius r?

|c|+r (at the point on the line through 0 and c furthest from 0).

Further loci and regions in the Argand diagramEdexcel A-Level Further Maths: Flashcards

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What is the locus $|z-a|=r$?

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What is the locus $|z-a|=r$ ?: A circle with centre $a$ and radius $r$ .
What is the locus $|z-a|=|z-b|$ ?: The perpendicular bisector of the line joining $a$ and $b$ .
What is the locus $\arg(z-a)=\theta$ ?: A half-line from $a$ (excluding $a$ ) at angle $\theta$ to the positive real direction.
What is the locus $|z-a|=k|z-b|$ for $k\neq1$ ?: A circle (circle of Apollonius).
How do you find the Cartesian equation of $|z-a|=k|z-b|$ ?: Put $z=x+iy$ , square both sides and complete the square.
Centre and radius of $|z-1|=2|z-4|$ ?: Centre $5$ , radius $2$ .
What is $\arg\left(\frac{z-a}{z-b}\right)$ in terms of arguments?: $\arg(z-a)-\arg(z-b)$ , the angle at $z$ subtended by $a$ and $b$ .
Locus of $\arg\left(\frac{z-a}{z-b}\right)=\beta$ ?: An arc of a circle through $a$ and $b$ (end points excluded).
Which arc if $\beta=\frac{\pi}{2}$ ?: A semicircle with diameter from $a$ to $b$ .
Region $p\leq\mathrm{Re}(z)\leq q$ ?: A vertical strip between $x=p$ and $x=q$ .
Region $\alpha\leq\arg(z-z_1)\leq\beta$ ?: The wedge between two half-lines from $z_1$ .
Greatest $|z|$ on a circle centre $c$ radius $r$ ?: $|c|+r$ (at the point on the line through $0$ and $c$ furthest from $0$ ).

Exam questions on Further loci and regions in the Argand diagram

The locus of points $z$ in the Argand diagram satisfying $|z-1|=2|z-4|$ is a circle $C$ .
Find the greatest value of $|z|$ for points $z$ on $C$ .2 marks
The region $R$ of the Argand diagram is defined by $\frac{\pi}{6}\leq\arg(z-1)\leq\frac{\pi}{3}$ and $\mathrm{Re}(z)\leq3$ .
Find the exact area of $R$ .2 marks
The locus $L$ is given by $\arg\left(\frac{z-2}{z+2}\right)=\frac{\pi}{2}$ .
Show that $L$ is part of the circle $x^2+y^2=4$ , stating which part, where $z=x+iy$ .3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).