Question

Find the maximum value of $|z|$ when $|z-\frac{3}{z}|=2$, $z$ being a complex number.

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

Answer 1

The correct option is B $3$

Since, $|z_1|-|z_2|\le |z_1-z_2|$
$\therefore |z|-\frac{3}{|z|}\le 2$
$\Rightarrow |z{|}^2-2|z|-3\le 0$
$\Rightarrow (|z|-3)(|z|+1)\le 0$
$\Rightarrow |z|\in [0,3][\because |z|\ge 0]$
$\Rightarrow |z{|}_{max}=3$