2022HSCShort answer

Q15 (d)

3 marks

Question

The complex number $z$ satisfies
$\left|z-\frac4z\right|=2.$

Using the triangle inequality, or otherwise, show that
$|z|\le\sqrt5+1.$

Try the question before revealing the answer.

Solution

Criteria	Marks
Provides correct solution	3
Uses the triangle inequality to obtain a relevant inequality involving only $	z
Correctly uses the triangle inequality, or equivalent merit	1

Sample answer:
By triangle inequality,

|z|\le\left|z-\frac4z\right|+\left|\frac4z\right|

$|z|\le2+\frac4{|z|}$
$|z|^2\le2|z|+4$
$|z|^2-2|z|-4\le0$

|z|\le\frac{2+\sqrt{4+4\times4}}2=\frac{2+\sqrt{20}}2=1+\sqrt5

Hence $|z|\le1+\sqrt5$ .

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Syllabus alignment

ProofComplex Numbers

The Nature of ProofUse formal proof language, contradiction, counter-examples and rigorous arguments involving numbers and inequalities.
Geometrical implications of complex numbersInterpret complex-number operations geometrically and solve problems involving roots and loci in the complex plane.

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