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Learning objectives
Core idea
Key formulae
Worked examples
Explore the graph
Common mistakes
Exam tip
Guided practice
Practice questions
Challenge
Summary
Quick quiz
Same topic, different board
Past paper questions

Learn9FM0Option · Further Pure Mathematics 2Complex numbersLoci and transformations

Complex numbers · Subtopic 1 of 1

Loci and Transformations of the Complex Plane

$|z-a|=r$ is a circle and $|z-a|=|z-b|$ a perpendicular bisector; $|z-a|=k|z-b|$ ( $k\neq1$ ) is a circle of Apollonius. A function $w=f(z)$ maps the $z$ -plane to the $w$ -plane: write $z$ in terms of $w$ , substitute into the locus, and simplify.

Edexcel 9FM0 Option≈ 45 minRevision 1 · updated 2026-10-27

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Section 1 of 14

Learning objectives

Sketch and find Cartesian equations of loci such as $|z-a|=k|z-b|$ .
Find images of lines and circles under $w=az+b$ and $w=\frac1z$ .
Use "make $z$ the subject" to transform a locus.

Pause and think

What does multiplying every complex number by $2i$ do to the plane?

Section 2 of 14

Core idea

Start hereFolding and stretching the plane

Think of the complex plane as a rubber sheet. A function $w=f(z)$ picks up every point $z$ and drops it at $w$ . Some functions just slide, turn and stretch the sheet; others, like $w=\frac1z$ , turn it inside out. Yet lines and circles always land as lines and circles.

The big idea

Loci: $|z-a|=r$ circle; $|z-a|=|z-b|$ bisector; $|z-a|=k|z-b|$ ( $k\ne1$ ) circle. To transform a locus under $w=f(z)$ : express $z$ in terms of $w$ and substitute.

1. Loci

|z-a|=k|z-b|

is a circle

$|z-3|=2|z|$
Example with $a=3$ , $b=0$ , $k=2$ .
$(x-3)^2+y^2=4(x^2+y^2)$
Square both sides with $z=x+iy$ .
$3x^2+6x+3y^2=9$
Collect.
$(x+1)^2+y^2=4$
Divide by 3; complete the square.

Centre

-1

, radius

2

. The

x^2

and

y^2

coefficients always match, so it is a circle unless

k=1

(then they cancel: a line).

2. Transformations

w=az+b:\ \text{enlarge by }|a|,\ \text{rotate by }\arg a,\ \text{translate by }b

$w=u+iv$: point in the image plane.

Worked example $w=iz+1$ on $|z-2|=3$

Find the image.

Goal: Image circle.
Strategy: Linear map: transform the centre, scale the radius.

$z=\frac{w-1}i$
Make $z$ the subject.
$\left|\frac{w-1}i-2\right|=3\Rightarrow|w-1-2i|=3|i|$
Multiply inside by $i$ .
$|w-(1+2i)|=3$
$|i|=1$ .

AnswerCircle centre

1+2i

, radius

3

✓ CheckCentre

2\mapsto i(2)+1=1+2i

✓.

w=\frac1z

on the line

\operatorname{Re}z=1

$z=\frac1w=\frac{u-iv}{u^2+v^2}$
Invert and realise the denominator.
$\operatorname{Re}z=\frac u{u^2+v^2}=1$
Apply the locus.
$u^2-u+v^2=0\Rightarrow\left(u-\tfrac12\right)^2+v^2=\tfrac14$
Complete the square.

A circle through the origin, centre

\frac12

, radius

\frac12

, with

w=0

itself excluded (no

z

maps there).

Transforming a locus

Drawing diagram…

The same routine works for every map in the specification.

Common misconceptionWrong direction Don't substitute

w=f(z)

into the

z

-locus; substitute

z

in terms of

w

Pause and think

Where does the unit circle go under $w=\frac1z$ ?

Summary

Loci: circles and bisectors.
$w=az+b$ : geometry.
$w=\frac1z$ : lines ↔ circles.
State excluded points.

Locus: Set of points satisfying a condition.
Image: Where a set is sent by the map.

HistoryMöbius transformations. August Möbius (of the strip) studied maps

w=\frac{az+b}{cz+d}

in the 1850s. They send circles and lines to circles and lines.

Fun factInside out.

w=\frac1z

turns the plane inside out: points near

0

go far away, and the unit circle maps to itself.

Section 3 of 14

Key formulae

Circle and bisector

|z-a|=r;\qquad|z-a|=|z-b|

Apollonius

|z-a|=k|z-b|,\ k\neq1\ \Rightarrow\ \text{circle}

Method

w=f(z)\ \Rightarrow\ z=f^{-1}(w)\ \Rightarrow\ \text{substitute into the }z\text{-locus}

TrickLinear maps are easy.

w=az+b

is an enlargement by

|a|

, rotation by

\arg a

and translation by

b

. A circle centre

c

, radius

r

maps to centre

ac+b

, radius

|a|r

Section 4 of 14

Worked examples

Example 1 · Apollonius circle0 / 3 steps

Find the locus

|z-3|=2|z|

Example 2 · w = 1/z0 / 3 steps

Find the image of the line

\operatorname{Re}z=1

under

w=\frac1z

Section 5 of 14

Explore the graph

Circle of Apollonius

Loading the interactive graph…

Try this

The circle is the locus $|z-c|=k|z|$ : points $k$ times as far from $c$ as from $0$ .
As $k\to1$ the circle grows enormous and tends to the perpendicular bisector.

Centre $-\frac{c}{k^2-1}$ , radius $\frac{kc}{k^2-1}$ .

In the real worldAerofoils. The Joukowsky map

w=z+\frac1z

turns a circle into a wing-shaped curve, which early aeronautical engineers used to calculate lift.

Section 6 of 14

Common mistakes

IncorrectSubstituting

w

into the

z

equation directly

CorrectMake

z

the subject first

The known locus is in

z

; you need it in terms of

w

IncorrectForgetting excluded points

Correct

w=\frac1z

never equals

0

Image circles through

0

have that point missing.

Incorrect

|z-3|=2|z|

is a bisector

CorrectIt's a circle

Only

k=1

gives a straight line.

A perfectly normal straight line

w = 1/z: congratulations, you're a circle now.

Section 7 of 14

Exam tip

Exam tipGive centre and radius. Write image circles as

|w-c|=r

(u-a)^2+(v-b)^2=r^2

, and state any excluded point.

Section 8 of 14

Guided practice

SA-FP-CT-G011 markL1 FoundationNon-calculatorTransformations

Under

w=2z+3i

, the circle

|z|=1

maps to a circle. Find its radius.

Salam Academy Original Practice · Proposed mark scheme

Section 9 of 14

Practice questions

SA-FP-CT-0012 marksL2 StandardNon-calculatorTransformations

Under

w=\frac1z

, the circle

|z|=2

maps to

|w|=r

. Find

r

Salam Academy Original Practice · Proposed mark scheme

SA-FP-CT-0022 marksL2 StandardNon-calculatorLoci

The locus

|z-2|=|z+2i|

is the line

y=mx

. Find

m

Salam Academy Original Practice · Proposed mark scheme

SA-FP-CT-0033 marksL3 ExamNon-calculatorLoci

The locus

|z-3|=2|z|

is a circle. Find its radius.

Salam Academy Original Practice · Proposed mark scheme

SA-FP-CT-0043 marksL3 ExamNon-calculatorTransformations

Under

w=\frac1z

, the line

\operatorname{Re}z=1

maps to a circle. Find its radius.

Salam Academy Original Practice · Proposed mark scheme

More questions on every skill in this subtopic are in the question bank under Practise.

Section 10 of 14

Challenge

SA-FP-CT-0053 marksL4 ChallengingNon-calculatorTransformations

Under

w=iz+1

, the circle

|z-2|=3

maps to

|w-c|=3

. Find the imaginary part of

c

Salam Academy Original Practice · Proposed mark scheme

Section 11 of 14

Summary

$|z-a|=k|z-b|$ : circle ( $k\neq1$ ) or bisector ( $k=1$ ).
Linear maps: enlarge, rotate, translate.
$w=\frac1z$ : lines and circles to lines and circles.
Method: $z$ in terms of $w$ , substitute.

Section 12 of 14

Quick quiz

Q1 · SA-FP-CT-Q011 markL1 FoundationNon-calculatorLoci

The locus

|z-1|=|z-5|

Salam Academy Original Practice · Proposed mark scheme

Section 13 of 14

Same topic, different board

Specification: A level Further Mathematics 9FM0
Where: Further Pure 2 · Complex numbers
Expected: Loci including |z − a| = k|z − b|; transformations w = az + b, w = 1/z and w = (az + b)/(cz + d), finding images of lines and circles.
Calculator: Permitted; show enough working to earn the method marks.
Typical style: Edexcel often opens with a routine part and builds to a "show that" or "prove" part.
Open in: 9FM0 · Option

Section 14 of 14

Past paper questions

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Salam Academy original content · Content status: reviewed and published. Syllabus mapping: Pearson Edexcel 9FM0, First teaching 2017 (official specification).