Complex Numbers And Quadratic Equations question 102
Question: If $ |z_1|=|z_2| $ and $ arg( \frac{z_1}{z_2} )=\pi $ , then $ z_1+z_2 $ is equal to
Options:
A) 0
B) Purely imaginary
C) Purely real
D) None of these
Show Answer
Answer:
Correct Answer: A
Solution:
We have $ arg( \frac{z_1}{z_2} )=\pi $
Þ $ arg(z_1)-arg(z_2)=\pi $
Þ $ arg(z_1)=arg(z_2)+\pi $ Let $ arg(z_2)=\theta $ , then $ arg $ $ (z_1)=\pi +\theta $ \ $ z_1=|z_1|[\cos (\pi +\theta )+i\sin (\pi +\theta )] $ $ =|z_1|(-\cos \theta -i\sin \theta ) $ and $ z_2=|z_2|(\cos \theta +i\sin \theta ) $ $ =|z_1|(\cos \theta +i\sin \theta ) $ $ (\because |z_1|=|z_2|) $ Hence $ z_1+z_2=0 $ .
BETA
