Question

If $\left|z\right|\ge 5$ then the least value $\left|z+\frac{2}{z}\right|$ is

• A. $\frac{23}{5}$
• B. $\frac{24}{5}$
• C. $5$
• D. none of these

Answer 1

The correct option is A $\frac{23}{5}$

$|z+\frac{2}{z}|\ge |z|-\left|\frac{2}{z}\right|$ using property $|z_1+z_2|\ge |z_1|-|z_2|$

So, given $|z|\ge 5$

$\Rightarrow |z+\frac{2}{z}|\ge 5-\frac{2}{5}$

$\ge \frac{25-2}{5}$

$\ge \frac{23}{5}$

So option $A$ is correct