JEE PYQ: Complex Numbers Question 17
Question 17 - 2020 (03 Sep Shift 2)
If $z_1, z_2$ are complex numbers such that $\text{Re}(z_1) = |z_1 - 1|$, $\text{Re}(z_2) = |z_2 - 1|$ and $\arg(z_1 - z_2) = \frac{\pi}{6}$, then $\text{Im}(z_1 + z_2)$ is equal to:
(1) $\frac{2}{\sqrt{3}}$
(2) $2\sqrt{3}$
(3) $\frac{\sqrt{3}}{2}$
(4) $\frac{1}{\sqrt{3}}$
Show Answer
Answer: (2)
Solution
$\text{Re}(z) = |z-1|$ gives $x = \sqrt{(x-1)^2+y^2}$, so $y^2 = 2x - 1$. Both $z_1, z_2$ lie on this parabola. With $\arg(z_1-z_2) = \pi/6$ and the parabola relation: $\text{Im}(z_1+z_2) = y_1 + y_2 = 2\sqrt{3}$.
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