Question

If $z_1$ and $z_2$ are two non-zero complex number such that $\left|\frac{z_1}{z_2}\right|$ = 2 and $\arg \left(z_1{z}_2\right)=\frac{3\pi }{2}$ , then $\frac{\overline{z_1}}{z_2}$ is equal to

• A. 2i
• B. -2
• C. -2i
• D. 2

Answer 1

The correct option is A 2i

$Let{z}_1=2r{e}^{i{\theta }_1},z_2=r{e}^{i{\theta }_2}arg\left(z_1{z}_2\right)={\theta }_1+{\theta }_2=\frac{3\pi }{2}\overline{z_1}=2r{e}^{-i{\theta }_1}\frac{\overline{z_1}}{z_2}=\frac{2r{e}^{-i{\theta }_1}}{r{e}^{i{\theta }_2}}=2r{e}^{-i({\theta }_1+{\theta }_2)}=2r{e}^{-i\frac{3\pi }{2}}=2i$