Question

If $|z-\frac{2}{z}|=1$, then the greatest value of $\left|z\right|$ is

• A. 2
• B. 1
• C. 4
• D. 3

Answer 1

The correct option is A 2

$|z-\frac{2}{z}|=1$ ...(1)
Let $\frac{2}{z}=w$
$\left|z\right|=|(z-w)+w|\le |z-w|+\left|w\right|$ ..(Triangle inequality)
$\Rightarrow \left|z\right|-\left|w\right|\le |z-w|$

$\Rightarrow \left|z\right|-\left|\frac{2}{z}\right|\le |z-\frac{2}{z}|$

$\Rightarrow \left|z\right|-\left|\frac{2}{z}\right|\le 1$ ...{ from 1 }

$\Rightarrow {\left|z\right|}^2-\left|z\right|-2\le 0\Rightarrow -1\le \left|z\right|\le 2\Rightarrow 0\le \left|z\right|\le 2$

Therefore, maximum value of $\left|z\right|$ is 2
Hence, option A is correct.