JEE PYQ: Complex Numbers Question 2
Question 2 - 2021 (16 Mar Shift 1)
Let $z$ and $w$ be two complex numbers such that $w = z\bar{z} - 2z + 2$, $\left|\frac{z+i}{z-3i}\right| = 1$ and $\text{Re}(w)$ has minimum value. Then, the minimum value of $n \in \mathbb{N}$ for which $w^n$ is real, is equal to ______.
Show Answer
Answer: 4
Solution
$|z+i| = |z-3i|$ implies $z = x + i$ for $x \in \mathbb{R}$. Then $w = x^2 - 2x + 3 - 2i = (x-1)^2 + 2 - 2i$. $\text{Re}(w)$ is minimum when $x = 1$: $w = 2 - 2i = 2\sqrt{2}e^{-i\pi/4}$. $w^n$ is real when $n\pi/4 = k\pi$, minimum $n = 4$.
BETA
