Question

If $\left| z - \frac{2}{z} \right|$ = 2, then the greatest value of | z | is

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

Answer 1

Answer: (D)

$\sqrt{5}$ + 1

Answer 2

Sol. We have | z | = $\left| z - \frac{4}{z} + \frac{4}{z} \right|$ £ $\left| z - \frac{4}{z} \right|$ + $\frac{4}{|z|}$

= 2 + $\frac{4}{|z|}$

Ž | z |² £ 2 | z | + 4 Ž (| z | – 1)² £ 5

Ž | z | – 1 £ $\sqrt{5}$ Ž | z | £ $\sqrt{5}$ + 1

Also, for z = $\sqrt{5}$ + 1

$\left| z - \frac{4}{z} \right|$ = 2

Therefore, the greatest value of | z | is$\sqrt{5}$ + 1.