Question

Let $z_1$ and $z_2$ are two complex numbers such that $(1-i)z_1=2z_2$ and $arg\left(z_1{z}_2\right)=\frac{\pi }{2}$ then $arg\left(z_2\right)$ is equals to:

• A. $\frac{3\pi }{8}$
• B. $\frac{\pi }{8}$
• C. $\frac{5\pi }{8}$
• D. $\frac{-7\pi }{8}$

Answer 1

The correct option is B $\frac{\pi }{8}$

$(1-i)z_1=2z_2$

$\frac{z_2}{z_1}=\frac{1}{2}-\frac{i}{2}$

Let $z_1=r_1{e}^{i{\theta }_1}and{z}_{=}{r}_1{r}^{{\theta }_2}$

$arg\left(\frac{z_2}{z_1}\right)={\tan }^{-1}\frac{-1/2}{1/2}=-\pi /4$

${\theta }_2-{\theta }_1=-\pi /4$ and $arg\left(z_1{z}_2\right)=\pi /2$ (given)

$⟹{\theta }_2-{\theta }_1=-\pi /4$ and ${\theta }_2+{\theta }_1=\pi /2$

$⟹2{\theta }_2=\pi /4,{\theta }_2=\pi /8$

$⟹arg\left(z_2\right)=\pi /8$