Complex Numbers

10 methods · 20 questions

★ Signature method unlocked: the technique this chapter rewards most often, with 3 worked questions.

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To identify the image curve of a constrained $z$ under a map $w = f (z)$ , either separate $w = u + i v$ into real and imaginary parts and eliminate the parameter, or carry $z = r e^{i θ}$ and read off $∣ w ∣$ and $ar g w$ . A real or imaginary part fixed gives a line; a fixed modulus or a Mobius map of a line or circle gives a circle or line.

Trigger: "find / sketch the locus of $w$ " where $w$ is a given function of a $z$ that runs along a line, a circle or an arc.

Instances:
(i) Split into $u, v$ and eliminate: lines $u =$ const or $v =$ const map to identifiable curves (often after a $tan (θ /2)$ substitution).
(ii) Mobius map $w = \frac{a z + b}{cz + d}$ sends lines and circles to lines and circles; test a few points or use the circle equation.
(iii) Translate an argument condition $ar g \frac{z - z _{1}}{z - z _{2}} = α$ into an arc of a circle, and a modulus condition $\frac{z - z _{1}}{z - z _{2}} = k$ into a circle or line.

Linked questions (3)

STEP 3 2011 · Q8

The complex numbers

z

and

w

are related by

w = \frac{1 + i z}{i + z} .

Let

z = x + i y

and

w = u + i v

, where

x

y

u

and

v

are real. Express

u

and

v

in terms of

x

and

y

By setting $x = tan (θ /2)$ , or otherwise, show that if the locus of $z$ is the real axis $y = 0$ , $- \infty < x < \infty$ , then the locus of $w$ is the circle $u^{2} + v^{2} = 1$ with one point omitted.
Find the locus of $w$ when the locus of $z$ is the line segment $y = 0$ , $- 1 < x < 1$ .
Find the locus of $w$ when the locus of $z$ is the line segment $x = 0$ , $- 1 < y < 1$ .
Find the locus of $w$ when the locus of $z$ is the line $y = 1$ , $- \infty < x < \infty$ .

STEP 3 2019 · Q6

6 The point

P

in the Argand diagram is represented by the complex number

z

, which satisfies

z z^{*} - a z^{*} - a^{*} z + a a^{*} - r^{2} = 0.

Here,

r

is a positive real number and

r^{2} \neq = a^{*} a

. By writing

∣ z - a ∣^{2}

(z - a) (z - a)^{*}

, show that the locus of

P

is a circle,

C

, the radius and the centre of which you should give.

(i) The point

Q

is represented by

ω

, and is related to

P

ω = \frac{1}{z}

. Let

C^{'}

be the locus of

Q

. Show that

C^{'}

is also a circle, and give its radius and centre.

If

C

and

C^{'}

are the same circle, show that

(∣ a ∣^{2} - r^{2})^{2} = 1

and that either

a

is real or

a

is imaginary. Give sketches to indicate the position of

C

in these two cases.

(ii) Suppose instead that the point

Q

is represented by

ω

, where

ω = \frac{1}{z ^{*}}

. If the locus of

Q

C

, is it the case that either

a

is real or

a

is imaginary?

STEP 2 2020 · Q7

7 In this question,

w = \frac{2}{z - 2}

.

(i) Let

z

be the complex number

3 + t i

, where

t \in R

. Show that

∣ w - 1∣

is independent of

t

. Hence show that, if

z

is a complex number on the line

Re (z) = 3

in the Argand diagram, then

w

lies on a circle in the Argand diagram with centre

1

.

Let

V

be the line

Re (z) = p

, where

p

is a real constant not equal to

2

. Show that, if

z

lies on

V

, then

w

lies on a circle whose centre and radius you should give in terms of

p

. For which

z

V

Im (w) > 0

?

(ii) Let

H

be the line

Im (z) = q

, where

q

is a non-zero real constant. Show that, if

z

lies on

H

, then

w

lies on a circle whose centre and radius you should give in terms of

q

. For which

z

H

Re (w) > 0