Question

If the complex number $z$ satisfies the condition $\left|z\right|\ge 3$ 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

Answer: 8

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