# If Z1, Z2, Z3 Are Complex Numbers Such that | Z 1 | = | Z 2 | = | Z 3 | = ∣ ∣ ∣ 1 Z 1 + 1 Z 2 + 1 Z 3 ∣ ∣ ∣ = 1 Then Find the Value of | Z 1 + Z 2 + Z 3 | . - Mathematics

If z1z2z3 are complex numbers such that $\left| z_1 \right| = \left| z_2 \right| = \left| z_3 \right| = \left| \frac{1}{z_1} + \frac{1}{z_2} + \frac{1}{z_3} \right| = 1$ then find the value of $\left| z_1 + z_2 + z_3 \right|$ .

#### Solution

$\left| z_1 + z_2 + z_3 \right| = \left| \frac{z_1 \bar{{z_1}}}{\bar{{z_1}}} + \frac{z_2 \bar{{z_2}}}{\bar{{z_2}}} + \frac{z_3 \bar{{z_3}}}{\bar{{z_3}}} \right|$

$= \left| \frac{\left| z_1 \right|^2}{\bar{{z_1}}} + \frac{\left| z_2 \right|^2}{\bar{{z_2}}} + \frac{\left| z_3 \right|^2}{\bar{{z_3}}} \right|$

$= \left| \frac{1}{\bar{{z_1}}} + \frac{1}{\bar{{z_2}}} + \frac{1}{\bar{{z_3}}} \right| [ \because \left| z_1 \right| = \left| z_2 \right| = \left| z_3 \right| = 1]$

$= \bar{\left| \frac{1}{z_1} + \frac{1}{z_2} + \frac{1}{z_3} \right|}$

$= 1 \left[ \because \left| \frac{1}{z_1} + \frac{1}{z_2} + \frac{1}{z_3} \right| = 1 \right]$

Thus, the value of $\left| z_1 + z_2 + z_3 \right|$ is 1.

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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 13 Complex Numbers
Exercise 13.2 | Q 25 | Page 33