Complex Numbers And Quadratic Equations question 650
Question: For the complex numbers $ z_1 $ and $ z_2 $ if $ |1-{{\bar{z}}_1}z_2{{|}^{2}}-|z_1-z_2{{|}^{2}}=k(1-|z_1{{|}^{2}})(1-|z_2{{|}^{2}}) $ then ?k? equals to
Options:
A) 1
B) $ -1 $
C) 2
D) $ -2 $
Show Answer
Answer:
Correct Answer: A
Solution:
$ {{| 1-{{{\bar{z}}}_1}z_2 |}^{2}}-{{| z_1-z_2 |}^{2}} $ $ =(1-{{\bar{z}}_1}z_2)(1-z_1{{\bar{z}}_2})-(z_1-z_2)({{\bar{z}}_1}-{{\bar{z}}_2}) $ $ =1-z_1{{\bar{z}}_2}-{{\bar{z}}_1}z_2+z_1{{\bar{z}}_1}z_2{{\bar{z}}_2}-(z_1{{\bar{z}}_1}-z_1{{\bar{z}}_2}-{{\bar{z}}_1}z_2+z_2{{\bar{z}}_2}) $ $ =1+z_1{{\bar{z}}_1}z_2{{\bar{z}}_2}-z_1{{\bar{z}}_1}-z_2{{\bar{z}}_2} $ $ =1+{{| z_1 |}^{2}}{{| z_2 |}^{2}}-{{| z_1 |}^{2}}-{{| z_2 |}^{2}} $ $ =(1-{{| z_1 |}^{2}})(1-{{| z_2 |}^{2}}) $ -
$ \Rightarrow k=1 $
BETA
