Question

The maximum value of $|z|$ when the complex number $z$ satisfies the condition $\left|z+\frac{2}{z}\right|$ = 2 is

• A. $\sqrt{3}$
• B. $\sqrt{3}+\sqrt{2}$
• C. $\sqrt{3}+1$
• D. $\sqrt{3}-1$

Answer 1

Answer: (C)

We have the identity,
$|z|=\left|z+\frac{2}{z}-\frac{2}{z}\right|\le \left|z+\frac{2}{z}\right|+\left|\frac{2}{z}\right|$
$\Rightarrow |z|\le 2+\frac{2}{|z|}$
$\Rightarrow |z{|}^2\le 2|z|+2$
$\Rightarrow |z{|}^2-2|z|+1\le 3$
$\Rightarrow (|z|-1{)}^2\le 3$
$\Rightarrow -\sqrt{3}\le |z|-1\le \sqrt{3}$
$\Rightarrow -\sqrt{3}+1\le |z|\le \sqrt{3}+1$
Hence, the maximum value $=\sqrt{3}+1$