Let S = \ zin mathbb C :z 2 +4z + 16 = 0\ . Then sum zin S z + √(3) i 2 is equal to :

Mathematics · Complex Numbers

JEE Main 2026 — 4 April, Evening Shift — Question 26

Let $S = \{z\in \mathbb{C}:z^2 +4z + 16 = 0\}$ . Then $\sum_{z\in S}\left|z + \sqrt{3} i\right|^2$ is equal to :

Option A:
42
Option B:
23
Option C:
27
Option D:
38
Correct

Answer: D

Step-by-step solution

$z^{2}+4 z+16=0$ $\Rightarrow(\mathrm{z}+2)^{2}=-12$ $\Rightarrow \mathrm{z}=-2 \pm 2 \sqrt{3} \mathrm{i}$ $\Rightarrow \mathrm{S}=\{-2+2 \sqrt{3} \mathrm{i},-2-2 \sqrt{3} \mathrm{i}\}$ $\therefore \sum_{\mathrm{z} \in \mathrm{S}}|\mathrm{z}+\sqrt{3} \mathrm{i}|^{2}$ $=|-2+2 \sqrt{3} i+\sqrt{3} i|^{2}+|-2-2 \sqrt{3} i+\sqrt{3} i|^{2}$ $=|-2+3 \sqrt{3} \mathrm{i}|^{2}+|-2-\sqrt{3} \mathrm{i}|^{2}$ $=38$

Answer key and solution verified before publishing.

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Exam: JEE Main 2026
Paper: 4 April, Evening Shift
Subject: Mathematics
Chapter: Complex Numbers
Topic: Introduction to Complex Numbers