JEE PYQ: Complex Numbers Question 34
Question 34 - 2019 (12 Apr Shift 2)
Let $z \in \mathbb{C}$ with $\text{Im}(z) = 10$ and it satisfies $\frac{2z-n}{2z+n} = 2i-1$ for some natural number $n$. Then:
(1) $n = 20$ and $\text{Re}(z) = -10$
(2) $n = 40$ and $\text{Re}(z) = 10$
(3) $n = 40$ and $\text{Re}(z) = -10$
(4) $n = 20$ and $\text{Re}(z) = 10$
Show Answer
Answer: (3)
Solution
$2z - n = (2i-1)(2z+n) = (2i-1) \cdot 2z + (2i-1)n$. $z(2 - 2(2i-1)) = n(2i-1+1) = 2ni$. $z(4-4i) = 2ni \Rightarrow z = \frac{ni}{2-2i} = \frac{ni(2+2i)}{8} = \frac{n(-2+2i)}{8} = \frac{n(-1+i)}{4}$. $\text{Im}(z) = \frac{n}{4} = 10 \Rightarrow n = 40$. $\text{Re}(z) = -10$.
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