Question

If the complex number $z$ satisfies the condition $\left|z\right|\geq 3$ and the least value of $\left|z + \frac{1}{z}\right|$ is $\frac{\lambda }{3}$ , then the value of $\lambda $ is equal to

Answer 1

We have, $\left|z\right|\geq 3$
By using inequality $\left|z + \frac{1}{z}\right|\geq \left|\left|\right. z \left|\right. - \frac{1}{\left|\right. z \left|\right.}\right|$
The least value of occurs, when $\left|z\right|=3$
$\left|z + \frac{1}{z}\right|_{m i n}=3-\frac{1}{3}=\frac{8}{3}$