Question

If z₁, z₂, z₃ are the complex numbers, such that |z₁| = |z₂| = |z₃| = $\left| \frac{1}{z_{1}} + \frac{1}{z_{2}} + \frac{1}{z_{3}} \right|$ = 1, then

|z₁ + z₂ + z₃| is –

• A. Equal to 1
• B. Less than 1
• C. Greater than 1
• D. Equal to 3

Answer 1

Answer: (A)

Equal to 1

Answer 2

Sol. |z₁| = |z₂| = |z₃| = 1 Ž ${\bar{z}}_{1}$=$\frac{1}{z_{1}}$,${\bar{z}}_{2}$= $\frac{1}{z_{2}}$, ${\bar{z}}_{3}$ = $\frac{1}{z_{3}}$

$|\overline{z_{1} + z_{2} + z_{3}}|$=$\left| \frac{1}{z_{1}} + \frac{1}{z_{2}} + \frac{1}{z_{3}} \right|$ =1

Q |z­₁ + z₂ + z₃| =$|\overline{z_{1} + z_{2} + z_{3}}|$ $|{\bar{z}}_{1} + {\bar{z}}_{2} + {\bar{z}}_{3}|$

=$\left| \frac{1}{z_{1}} + \frac{1}{z_{2}} + \frac{1}{z_{3}} \right|$ = 1 (given)