JEE Main 2021 (Online) 27th August Evening ShiftQuestion 1
Let $z_1$ and $z_2$ be two complex numbers such that $\arg(z_1-z_2)=\frac{\pi}{4}$ and $z_1,z_2$ satisfy the equation $|z-3|=\operatorname{Re}(z)$. Then the imaginary part of $z_1+z_2$ is equal to $\_\_\_\_\_\_$ .
Complex Numbers
Answer$6$
Solution
$|z-3|=\mathrm{Re}z\Rightarrow(x-3)^2+y^2=x^2\Rightarrow y^2=6x-9$. For points $(x_k,y_k)$: $x_k=\frac{y_k^2+9}6$. Slope of $z_1z_2$ is $\tan\frac\pi4=1$: $\dfrac{y_1-y_2}{x_1-x_2}=\dfrac{6}{y_1+y_2}=1\Rightarrow y_1+y_2=6$. Imaginary part of $z_1+z_2$ is 6.