Complex Numbers And Quadratic Equations question 711
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
Options:
A) equal to 1
B) less than 1
C) greater than 3
D) equal to 3
Show Answer
Answer:
Correct Answer: A
Solution:
$ | z_1 |=| z_2 |=| z_3 |=1 $ (given) Now, $ | z_1 |=1\Rightarrow {{| z_1 |}^{2}}=1\Rightarrow z_1{{\bar{z}}_1}=1 $ Similarly, $ {z_2}{{\bar{z}}_2}=1,z_3{{\bar{z}}_3}=1 $ Now, $ | \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 |\overline{z_1+z_2+z_3}|=1\Rightarrow |z_1+z_2+z_3|=1 $
BETA
