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