Question

Let $z_1$ and $z_2$ be two complex numbers such that $\overline{z_1}+i\overline{z_2}=0,\arg \left(z_1{z}_2\right)=\pi .\text{Then}$ 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)

The correct options are
A $\arg z_1\in (0,\frac{5\pi }{6})$
D $\arg z_2\in (0,\frac{\pi }{2})$

$\overline{z_1}+i\overline{z_2}=0$
$\Rightarrow \overline{z_1}=-i\overline{z_2}$
$\Rightarrow \overline{\overline{z_1}}=\overline{-i\overline{z_2}}$
$\Rightarrow z_1=i{z}_2$
$\Rightarrow \arg \left(z_1\right)=\arg \left(i{z}_2\right)$
$\Rightarrow \arg z_1=\arg i+\arg z_2$
$\Rightarrow \arg z_1-\arg z_2=\frac{\pi }{2}\dots \left(1\right)$
$\because \arg z_1+\arg z_2=\pi \dots \left(2\right)$
By equations $\left(1\right)$ and $\left(2\right)$
$2\arg z_1=\frac{3\pi }{2}\Rightarrow \arg z_1=\frac{3\pi }{4}$
and $\arg z_2=\frac{\pi }{4}$