Question

Let $z_1$ and $z_2$ be two complex numbers such that $\overline{z_1}+i\overline{z_2}=0,\arg \left(z_1{z}_2\right)=\pi$. Then which of the following is/are ture?

• A. $\arg z_1\in (0,\frac{5\pi }{6})$
• B. $\arg z_1\in (0,\frac{\pi }{2})$
• C. $\arg z_2\in (\frac{\pi }{3},\frac{2\pi }{3})$
• D. $\arg z_2\in (0,\frac{\pi }{2})$

Answer 1

Answer: (A), (D)

$\arg z_2\in (0,\frac{\pi }{2})$

$\overline{z_1}+i\overline{z_2}=0\Rightarrow \overline{z_1}=-i\overline{z_2}\Rightarrow \overline{\overline{z_1}}=\overline{-i\overline{z_2}}\Rightarrow z_1=i{z}_2\Rightarrow \arg \left(z_1\right)=\arg \left(i{z}_2\right)\Rightarrow \arg z_1=\arg i+\arg z_2\Rightarrow \arg z_1-\arg z_2=\frac{\pi }{2}\cdots \left(1\right)\arg z_1+\arg z_2=\pi \cdots \left(2\right)$

By equations $\left(1\right)$ and $\left(2\right)$, we get
$2\arg z_1=\frac{3\pi }{2}\Rightarrow \arg z_1=\frac{3\pi }{4}\text{and}\arg z_2=\frac{\pi }{4}$