Question

$\text{If}|z-\frac{4}{z}|=2,\text{then the greatest value of}|z|\text{is}$

• A. $\sqrt{5}+1$
• B. $\sqrt{3}+1$
• C. $2+\sqrt{2}$
• D. $1+\sqrt{2}$

Answer 1

The correct option is A

$\sqrt{5}+1$

$\text{Given:}|z-\frac{4}{z}|=2$

We have,
$|z|=|z-\frac{4}{z}+\frac{4}{z}|\le |z-\frac{4}{z}|+\frac{4}{|z|}=2+\frac{4}{|z|}\Rightarrow |z{|}^2\le 2|z|+4\Rightarrow (|z|-1{)}^2\le 5\Rightarrow -\sqrt{5}+1\le |z|\le 1+\sqrt{5}$

Also, for $z=\sqrt{5}+1,|z-\frac{4}{z}|=2$
Therefore, the greatest value of $|z|$ is $\sqrt{5}+1$ which is attained when $z=\sqrt{5}+1$