Complex Numbers And Quadratic Equations question 82
Question: Let $ z_1 $ be a complex number with $ |z_1|=1 $ and $ z_2 $ be any complex number, then $ | \frac{z_1-z_2}{1-z_1{{{\bar{z}}}_2}} |= $ [Orissa JEE 2004]
Options:
A) 0
B) 1
C) - 1
D) 2
Show Answer
Answer:
Correct Answer: B
Solution:
We have $ |z_1|\ =1 $ and $ z_2 $ be any complex number.
$ \Rightarrow | \ \frac{z_1-z_2}{1-z_1{{{\bar{z}}}_2}} |\ =\frac{|z_1-z_2|}{| \ 1-\frac{{{{\bar{z}}}_2}}{{{{\bar{z}}}_1}}\ |} $ ; $ \because \ z_1{{\bar{z}}_1}=\ |z_1{{|}^{2}} $ $ =\frac{|z_1-z_2|}{|{{{\bar{z}}}_1}-{{{\bar{z}}}_2}|}|{{\bar{z}}_1}| $ ; Given that $ \because \ |{{\bar{z}}_1}|\ =1 $ $ =\frac{|z_1-z_2|}{|\overline{z_1-z_2}|}=\frac{|z_1-z_2|}{|z_1-z_2|}=1 $ .
BETA
