Complex Numbers And Quadratic Equations question 691
Question: If z in any complex number satisfying then which of the following is correct?
Options:
A) $ \arg (z-1)=2argz $
B) $ 2\arg (z)=\frac{2}{3}arg(z^{2}-z) $
C) $ \arg (z-1)=\arg (z+1) $
D) $ \arg z=2\arg (z+1) $
Show Answer
Answer:
Correct Answer: A
Solution:
Since $ | z-1 |= 1 \therefore z -1 = {e^{i\theta }} $ , where $ \arg | z-1 |=\theta $
$ \therefore z=1+cos\theta +i\sin \theta $ $ =2\cos \frac{\theta }{2}[ \cos \frac{\theta }{2}+i\sin \frac{\theta }{2} ] $ $ =2\cos \frac{\theta }{2}.{e^{i\theta /2}}=2{{\cos }^{2}}\frac{\theta }{2}+2i\sin \frac{\theta }{2}\cos \frac{\theta }{2} $ Thus, $ \arg ( z -1 ) = 2 arg z. $
BETA
