Question

Let $z_{1}$ and $z_{2}$ be two complex numbers such that $\bar{z}_{1}+i \bar{z}_{2}=0 $ and arg $\left(z_{1}z_{2}\right)=\pi$, then find arg $\left(z_{1}\right)$.

• A. $\pi$
• B. $\pi/4$
• C. $\pi/2$
• D. $3\pi/4$

Answer 1

Answer: (D)

Given that $\bar{z}_{1}+i \bar{z}_{2} =0$
$\Rightarrow \, z_{1}=iz_{2}$,
i.e., $z_{2}=-iz_{1}$
Thus arg $\left(z_{1}z_{2}\right)=arg\, z_{1}+arg\, z_{2}=arg \,z_{1}+arg \left(-iz_{1}\right)=\pi$
$\Rightarrow \, arg \left(-iz_{1}^{2}\right)=\pi $
$\Rightarrow \, arg \left(-i\right)+arg\left(z_{1}^{2}\right)=\pi$
$\Rightarrow \, arg \left(-i\right)+2\, arg \left(z_{1}\right)=\pi$
$\Rightarrow \, \frac{-\pi}{2}+2 \,arg \left(z_{1}\right)=\pi$
$\Rightarrow arg \left(z_{1}\right)=\frac{3\pi}{4}$