Complex Numbers And Quadratic Equations question 133
Question: For any two complex numbers $ z_1 $ and $ z_2 $ and any real numbers a and b; $ |(az_1-bz_2){{|}^{2}}+|(bz_1+az_2){{|}^{2}}= $ [IIT 1988]
Options:
A) $ (a^{2}+b^{2})(|z_1|+|z_2|) $
B) $ (a^{2}+b^{2})(|z_1{{|}^{2}}+|z_2{{|}^{2}}) $
C) $ (a^{2}+b^{2})(|z_1{{|}^{2}}-|z_2{{|}^{2}}) $
D) None of these
Show Answer
Answer:
Correct Answer: B
Solution:
$ |(az_1-bz_2){{|}^{2}}+|(bz_1+az_2){{|}^{2}} $ $ =a^{2}|z_1{{|}^{2}}+b^{2}|z_2{{|}^{2}}-2Re(ab)|z_1{{\overline{z}}_2}|+b^{2}|z_1{{|}^{2}}+ $ $ a^{2}|z_2{{|}^{2}}+2Re(ab)|{{\overline{z}}_1}z_2| $ $ =(a^{2}+b^{2})(|z_1{{|}^{2}}+|z_2{{|}^{2}}) $
BETA
