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4-1.Complex numbers
medium
Let $z _{1}$ and $z _{2}$ be two complex numbers such that $\overline{ z }_{1}=i \overline{ z }_{2}$ and $\arg \left(\frac{ z _{1}}{\overline{ z }_{2}}\right)=\pi$. Then
A
$\arg z _{2}=\frac{\pi}{4}$
B
$\arg z _{2}=-\frac{3 \pi}{4}$
C
$\arg z _{1}=\frac{\pi}{4}$
D
$\arg z _{1}=-\frac{3 \pi}{4}$
(JEE MAIN-2022)
Solution
$\overline{ z }_{1}= i _{2}$
$z _{1}=- iz _{2}$
$\arg \left(\frac{ z _{1}}{\overline{ z }_{2}}\right)=\pi$
$\arg \left(- i \frac{ z _{2}}{\overline{ z }_{2}}\right)=\pi \quad \arg \left( z _{2}\right)=\theta$
$-\frac{\pi}{2}+\theta+\theta=\pi$
$2 \theta=\frac{3 \pi}{2}$
$\arg \left( z _{2}\right)=\theta=\frac{3 \pi}{4}, \arg z _{1}=\frac{\pi}{4}$
Standard 11
Mathematics
