Question

If $z$ is a complex number such that $|z-\frac{4}{z}|=2$, then the greatest value of $|z|$ is

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

Answer 1

Answer: (D)

$1+\sqrt{5}$

$\Rightarrow |z|=|z-\frac{4}{z}+\frac{4}{z}|\le |z-\frac{4}{z}|+\left|\frac{4}{z}\right|$

$\Rightarrow |z|\le 2+\frac{4}{|z|}$

$\Rightarrow |z{|}^2-2|z|\le 4$

$\Rightarrow (|z|-1{)}^2\le 5$

$\Rightarrow |z|\le \sqrt{5}+1$