First slide

Algebra of complex numbers

Question

If $|z - \frac{4}{z}| = 2,$ then the maximum value of $|z|$ is equal to

Moderate

A

1
B

$2 + \sqrt{2}$
C

$\sqrt{3} + 1$
D

$\sqrt{5} + 1$

Solution

$\begin{aligned} | z | - \frac{4}{| z |} \leq | | z |- \frac{4}{| z |}| \leq |z - \frac{4}{z}| = 2 \\ \Rightarrow | z |^{2} - 2 | z | - 4 \leq 0 \\ \Rightarrow (| z | - 1)^{2} \leq 5 \Rightarrow | z | \leq 1 + \sqrt{5} . \end{aligned}$
The maximum value $\sqrt{5} + 1$ is attained when $z = \sqrt{5} + 1 .$

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