question_answer

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) \[\arg (z)=2\arg (z+1)\]

Correct Answer: A

Solution :

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,\[\arg (z-1)=2\arg (z)\]