Let S = \ z in mathbb C : frac z -6 i z -2 i =1 . and . frac z -8+2 i z +2 i =3/5 \ . Then sum z in s z 2 is equal to

Mathematics · Complex Numbers

JEE Main 2026 — 24 January, Morning Shift — Question 9

Let $\mathrm{S}=\left\{\mathrm{z} \in \mathbb{C}:\left|\frac{\mathrm{z}-6 \mathrm{i}}{\mathrm{z}-2 \mathrm{i}}\right|=1\right.$ and $\left.\left|\frac{\mathrm{z}-8+2 \mathrm{i}}{\mathrm{z}+2 \mathrm{i}}\right|=\frac{3}{5}\right\}$ . Then $\sum_{\mathrm{z} \in \mathrm{s}}|\mathrm{z}|^{2}$ is equal to

Option A:
$398$
Option B:
$413$
Option C:
$4423$
Option D:
$385$
Correct

Answer: D

Step-by-step solution

Solving $\left|\frac{z-6 i}{z-2 i}\right|=1 \Rightarrow y=4 \ldots \ldots$ (1) (where $\mathrm{z}=\mathrm{x}+\mathrm{iy}$ )

Now solving $\left|\frac{z-8+2 i}{z+2 i}\right|=\frac{3}{5}$

\begin{gathered} \Rightarrow \mathrm{x}^{2}+\mathrm{y}^{2}-25 \mathrm{x}+4 \mathrm{y}+104=0 \end{gathered}

Solving & ⇒ $\mathrm{z}=17+4 \mathrm{i} \& 8+4 \mathrm{i}$ $\Rightarrow \Sigma|z|^{2}=(17)^{2}+^{2}+(8)^{2}+^{2}=385$

Answer key and solution verified before publishing.

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Exam: JEE Main 2026
Paper: 24 January, Morning Shift
Subject: Mathematics
Chapter: Complex Numbers
Topic: Geometry of Complex Numbers