JEE Main 2020 (Online) 3rd September Evening ShiftQuestion 1
If $z_1,z_2$ are complex numbers such that $\mathrm{Re}(z_1)=|z_1-1|$, $\mathrm{Re}(z_2)=|z_2-1|$ and $\arg(z_1-z_2)=\frac\pi6$, then $\mathrm{Im}(z_1+z_2)$ is equal to :
Complex Numbers
Solution
$\mathrm{Re}(z)=|z-1|\Rightarrow x^2=(x-1)^2+y^2\Rightarrow y^2=2x-1$. $x_1-x_2=\frac{y_1^2-y_2^2}2$. Slope $\frac{y_1-y_2}{x_1-x_2}=\frac{2}{y_1+y_2}=\tan\frac\pi6=\frac1{\sqrt3}$. $\Rightarrow y_1+y_2=2\sqrt3$. Answer: (C).