First slide

Algebra of complex numbers

Question

If $z_{1}, z_{2} and z_{3}$ are complex numbers such that $|z_{1}| = |z_{2}| = |z_{3}| = |\frac{1}{z_{1}} + \frac{1}{z_{2}} + \frac{1}{z_{3}}| = 1,$ then $|z_{1} + z_{2} + z_{3}|$ is

Easy

A

equal to 1
B

less than 1
C

greater than 3
D

equal to 3

Solution

$|z_{1}| = |z_{2}| = |z_{3}| = 1$
$\begin{array}{l} now, |z_{1}| = 1 \Rightarrow {|z_{1}|}^{2} = 1 \\ \Rightarrow z_{1} {\bar{z}}_{1} = 1 \Rightarrow {\bar{z}}_{1} = \frac{1}{z_{1}} \end{array}$
Similarly, $z_{2} {\bar{z}}_{2} = 1 and z_{3} {\bar{z}}_{3} = 1$
$\begin{array}{l} again, |\frac{1}{z_{1}} + \frac{1}{z_{2}} + \frac{1}{z_{3}}| = 1 \\ \Rightarrow |{\bar{z}}_{1} + {\bar{z}}_{2} + {\bar{z}}_{3}| = 1 \\ \Rightarrow |z_{1} + z_{2} + z_{3}| = 1 \\ \Rightarrow |z_{1} + z_{2} + z_{3}| = 1 \end{array}$

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App