Complex numbers 2 — Question 16

QCAA 2024, Paper 2 · 6 marks

Q16 · 2024 · Technology-activeComplex familiar6 marks

QUESTION 16 (6 marks)

Two subsets of the complex plane are

S=\{z:|z-1|=4\}

and

T=\left\{z:\arg(z+i)=\frac{\pi}{3}\right\},

where

z\in\mathbb C

. Determine the complex number/s where

S

and

T

intersect. Leave your answer/s in Cartesian form. Provide an Argand diagram with a sketch of subsets

S

and

T

as part of your solution.

Question linkOriginal paper

WORKED SOLUTION

QCAA guide · typeset solution

6 marks

ANSWER

z\approx2.67+3.63i

Worked solution

On an Argand diagram,

S

is a circle of radius 4 centred at

(1,0)

and

T

is a ray starting at

(0,-1)

and moving through quadrants IV and I. Let

z=x+yi

. Then

S:(x-1)^2+y^2=16.

For

T

y=\tan\left(\frac{\pi}{3}\right)x-1=\sqrt3x-1,\qquad x>0.

Substituting into the circle equation gives

(x-1)^2+(\sqrt3x-1)^2=16.

A GDC gives

x\approx-1.31

x\approx2.67

. Only

x\approx2.67

lies on the required ray. Then

y\approx\sqrt3(2.67)-1\approx3.63.

Therefore the intersection is

z\approx2.67+3.63i.

correctly represents subset

S

on the Argand diagram as a circle of radius 4 centred at

(1,0)

[1 mark]

correctly represents subset

T

on the Argand diagram as a ray starting at

(0,-1)

and moving through quadrants IV and I

[1 mark]

determines an equation to represent subset

S

[1 mark]

determines an equation to represent subset

T

[1 mark]

determines the reasonable solution for

x

[1 mark]

determines where

S

and

T

intersect expressed as a complex number in Cartesian form

[1 mark]

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Complex numbers 2 — Question 16

QUESTION 16 (6 marks)

QCAA guide · typeset solution

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