Question

If the complex number $z$ satisfies the condition $|z|\ge 3$, then the least value of $|z+\frac{1}{z}|$ is equal to :

• A. $5/3$
• B. $8/3$
• C. $11/3$
• D. $noneofthese$

Answer 1

The correct option is B $8/3$

$\begin{array}{c}\left|z\right|\ge 3,\left|z+\frac{1}{z}\right|=?\\ \Rightarrow \frac{1}{\left|z\right|}\le \frac{1}{3}\\ \Rightarrow -\frac{1}{\left|z\right|}\ge -\frac{1}{3}\\ \therefore \left|z\right|-\frac{1}{\left|z\right|}\ge 3-\frac{1}{3}=\frac{8}{3}\to \left(ii\right)\left[\left|z_1=z_2\right|\ge \left|\left|z_1\right|-\left|z_2\right|\right|\right]\\ \therefore \left|z+\frac{1}{z}\right|\ge \left|\left|z\right|-\frac{1}{\left|z\right|}\right|\to \left(i\right)\\ Addingequation\left(i\right)and\left(ii\right)\\ \left|z+\frac{1}{z}\right|\ge \frac{8}{3}\end{array}$