Complex Numbers And Quadratic Equations question 139
Question: If $ z_1,z_2,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 [MP PET 2004; IIT Screening 2000]
Options:
A) Equal to 1
B) Less than 1
C) Greater than 3
D) Equal to 3
Show Answer
Answer:
Correct Answer: A
Solution:
$ 1=| \frac{1}{z_1}+\frac{1}{z_2}+\frac{1}{z_3} | $ $ =| \frac{z_1{{{\bar{z}}}_1}}{z_1}+\frac{z_2{{{\bar{z}}}_2}}{z_2}+\frac{z_3{{{\bar{z}}}_3}}{z_3} | $ $ (\because |z_1{{|}^{2}}=1=z_1{{\overline{z}}_1},etc) $ $ =|{{\bar{z}}_1}+{{\bar{z}}_2}+{{\bar{z}}_3}|=|\overline{z_1+z_2+z_3}|=|z_1+z_2+z_3| $ $ (\because |{{\bar{z}}_1}|=|z_1|) $
BETA
