Question

If $|z-1|=2,\arg (z-1)=\frac{\pi }{6}$ then $z$ is

• A. $\sqrt{3}+i$
• B. $\sqrt{3}+1+i$
• C. $\sqrt{3}-1+i$
• D. $1-\sqrt{3}i$

Answer 1

The correct option is B $\sqrt{3}+1+i$

From eulers form of complex number,
$z-1=|z-1|e^{i\arg (z-1)}$, where $|z-1|=2,\arg (z-1)=\frac{\pi }{6}$
$\therefore z-1=2e^{i\pi /6}$
$\Rightarrow z=2(\cos \frac{\pi }{6}+i\sin \frac{\pi }{6})+1$
$\Rightarrow z=\sqrt{3}+i+1$