Question

If $|z+\frac{6}{z}|=5,$ then greatest value of $\left|z\right|$ is equal to

• A. 6
• B. 5
• C. 1
• D. 3

Answer 1

The correct option is A 6

By triangle inequality, we have
$|\left|z\right|-\frac{6}{\left|z\right|}|\le |z+\frac{6}{z}|=5$
i.e. $|z|-\frac{6}{|z|}\le 5$
i.e. $|z|2-5|z|-6\le 0$
i.e. $(|z|-6)(|z|+1)\le 0$
i.e. $-1\le |z|\le 6$
Hence, the greatest value of $\left|z\right|$ is 6