Question

If $ z $ is any complex number satisfying $ |z-1|\,\,=1 $ , then which of the following is correct?

• A. $ \arg (z-1)=2\arg (z) $
• 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) $

Answer 1

Answer: (A)

Clearly, $ |z-1|\,\,=1 $ represents a circle with centre at $ (1,\,\,0) $ and radius 1.
Let $ P(z) $ be any point on it.
Then, $ \arg (z-1)=\angle XCP=\theta $ [say]
$ \therefore $ $ \arg (z)=\angle XOP=\frac{\theta }{2} $
Hence, $ \arg (z-1)=2\arg (z) $