JEE Main 2022 · 25 Jun · Shift 2, Mathematics question 3

Relation. $\bar z_1=i\bar z_2$ gives $z_1=-iz_2$, so $\dfrac{z_1}{\bar z_2}=-i\,e^{2i\theta_2}$ with $\theta_2=\arg z_2$.

Condition. $-\frac\pi2+2\theta_2\equiv\pi$, so $\theta_2=\frac{3\pi}4$ or $-\frac\pi4$, and $\arg z_1=\theta_2-\frac\pi2=\frac\pi4$ or $-\frac{3\pi}4$.

Options. A and B are never true. With $\arg z_2=\frac{3\pi}4$, $\arg z_1=\frac\pi4$, the keyed (C). Note: $\arg z_2=-\frac\pi4$, $\arg z_1=-\frac{3\pi}4$ also satisfies both conditions, so (D) is possible too; NTA's key gives (C) only.

(C)