Question

If $z_1$ and $z_2$ are two non-zero complex numbers 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

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

Answer 1

The correct option is A $2$

Let $z_1=a+ib$ and $z_2=c+id$

$\left|\frac{z_1}{z_2}\right|=2$

$\left|z_1\right|=2\left|z_2\right|$

arg($z_1{z}_2)=\frac{3\pi }{2}$

${\tan }^{-1}\frac{b}{a}+{\tan }^{-1}\frac{d}{c}=\frac{3\pi }{2}$

${\tan }^{-1}\frac{bd+ac}{ad-bc}=\frac{3\pi }{2}$

so,$ad=bc$

$\frac{b}{a}=\frac{d}{c}$

from $\left|z_1\right|=2\left|z_2\right|$

$a^2+b^2=4(c^2+d^2)$

$\frac{a}{c}$=2

so $\frac{z_1}{z_2}=\frac{a+ib}{c+id}=\left(\frac{a}{c}\right)\frac{1+i\frac{b}{a}}{1+i\frac{c}{d}}$

$\frac{a}{c}$=2

so option${}^{\prime }{A}^{\prime }$ is correct.