Question

If $\left|Z-\frac{4}{z}\right|=2$, then the maximum value of $\left|Z\right|$ is equal to

• A. $\sqrt{3}+1$
• B. $\sqrt{5}+1$
• C. $2$
• D. $2+\sqrt{2}$

Answer 1

Answer: (B)

$\left|Z\right|=\left|\left(Z-\frac{4}{Z}\right)+\frac{4}{Z}\right|\Rightarrow \left|Z\right|=\left|Z-\frac{4}{Z}+\frac{4}{Z}\right|$
$\Rightarrow \left|Z\right|\le \left|Z-\frac{4}{Z}\right|+\frac{4}{|Z|}\Rightarrow \left|Z\right|\le 2+\frac{4}{|Z|}$
$\Rightarrow {\left|Z\right|}^2-2\left|Z\right|-4\le 0$
$\left(\left|Z\right|-\left(\sqrt{5}+1\right)\right)\left(\left|Z\right|-\left(1-\sqrt{5}\right)\right)\le 0\Rightarrow 1-\sqrt{5}\le \left|Z\right|\le \sqrt{5}+1$