Question

If $z_1$ and $z_2$ are two complex numbers such that $\mid \frac{z_1}{z_2}\mid =2$ and $arg\left(z_1{z}_2\right)=\frac{3\pi }{2}$, and $\frac{\overline{z_1}}{z_2}$ is equal to:

• A. $2$
• B. $-2$
• C. $-2i$
• D. $2i$

Answer 1

Answer: (C)

$-2i$

$\frac{|z_1|}{|z_2|}=2⟹|z_1|=2|z_2|$

Let$|z_1|=r,|z_2|=2r$

$|z_1|=2r{e}^{i{\theta }_1}$

$|z_2|=r{e}^{i{\theta }_2}$

$arg\left(z_1{z}_2\right)={\theta }_1{\theta }_2=3\pi /2$

$\overline{z_1}=2r{e}^{-i{\theta }_1}$

$\frac{\overline{|z_1|}}{|z_2|}=\frac{2r{e}^{-i{\theta }_1}}{r{e}^{i{\theta }_2}}⟹2e^{-i({\theta }_1+{\theta }_2)}⟹2e^{-i3\pi /2}=-2i$