Question

Maximum distance from origin of the points z satisfying the relation |z + 1/z| = 1 is –

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

Answer 1

Answer: (A)

($\sqrt{5}$ + 1)/2

Answer 2

Sol. We may assume |z| ³ $\frac{1}{|z|}$ for otherwise, we may interchange z and 1/z in the given equation. We have

z| – $\frac{1}{|z|}$ = $\left| |z|–\frac{1}{|z|} \right|$

= $\left| |z|–\left| –\frac{1}{|z|} \right| \right|$£$\left| z–\left( - \frac{1}{z} \right) \right|$

= |z + 1/z| = 1

Thus, |z| – $\frac{1}{|z|}$£ 1 Ž |z|² – |z| – 1 £ 0

Ž |z| lies between the roots of

|z|² – |z| – 1 = 0 Ž $\frac{1}{2}$ (1 – $\sqrt{5}$) £ |z| £ $\frac{1}{2}$ (1 + $\sqrt{5}$)

As z ¹ 0 |z| > 0, therefore, 0 < |z| £ $\frac{1}{2}$ ($\sqrt{5}$ + 1)

Taking z = $\frac{i}{2}$ ($\sqrt{5}$ + 1), we get $\left| z + \frac{1}{z} \right|$ = 1.

Thus, maximum possible value of |z| is ($\sqrt{5}$ + 1)/2.