Question

Let $z_1$ and $z_2$ be two complex number such that $\overline{z_1}+i\overline{z_2}=0$ and $\arg \left(z_1{z}_2\right)=\pi$. then $\arg \left(z_1\right)$ is

• A. $\frac{\pi }{4}$
• B. $\frac{3\pi }{4}$
• C. $\frac{5\pi }{4}$
• D. $\frac{7\pi }{4}$

Answer 1

Answer: (A)

$\frac{\pi }{4}$

$z_1=x_1+i{y}_1$ $z_2=x_2+i{y}_2$

$\overline{z_1}=x_1-i{y}_1$ ${\overline{z}}_2=x_2-i{y}_2$

$x_1-i{y}_1+i{x}_2+y_2=0$

$x_1=y_2,y_1=x_2$

$\frac{y_1}{x_1}=\frac{-x_2}{y_2}$

If arg $\left(z_1{z}_2\right)=\pi$

$z_1{z}_2=x_1{x}_2-y_1{y}_2+i(x_1{y}_2+y_2{x}_1)\to x_1{y}_2=-y_1{x}_2$

From this we get $\frac{y_1}{x_1}=\frac{-y_1}{x_2}=\frac{x_1}{y_1}\to y_1=x_1$

$arg\left(z_1\right)=\frac{\pi }{4}$ A is correct.