Let z l , z 2 and z 3 be three complex numbers on the circle z =1 with arg (z 1 )=-π/4, arg (z 2 )=0 and arg (z 3 )=π/4 . If z 1 overline z 2 + z 2 overline z 3 + z 3 overline z 1 2 =α+β √(2), α, β in Z , then the value of α 2 +β 2 is:

Mathematics · Complex Numbers

JEE Main 2025 — 22 January, Morning Shift — Question 4

Let $\mathrm{z}_{\mathrm{l}}, \mathrm{z}_{2}$ and $\mathrm{z}_{3}$ be three complex numbers on the circle $|z|=1$ with $\arg \left(z_{1}\right)=\frac{-\pi}{4}, \arg \left(z_{2}\right)=0$ and $\arg \left(z_{3}\right)=\frac{\pi}{4}$ . If $\left|z_{1} \overline{\mathrm{z}}_{2}+\mathrm{z}_{2} \overline{\mathrm{z}}_{3}+\mathrm{z}_{3} \overline{\mathrm{z}}_{1}\right|^{2}=\alpha+\beta \sqrt{2}, \alpha, \beta \in \mathrm{Z}$ , then the value of $\alpha^{2}+\beta^{2}$ is:

Option A:
24
Option B:
41
Option C:
31
Option D:
29
Correct

Answer: D

Step-by-step solution

\begin{aligned} &z_1=e^{-i\pi/4},\qquad z_2=1,\qquad z_3=e^{i\pi/4},\\[6pt] &\overline{z_2}=1,\quad \overline{z_3}=e^{-i\pi/4}=z_1,\quad \overline{z_1}=e^{i\pi/4}=z_3.\\[6pt] &S=z_1\overline{z_2}+z_2\overline{z_3}+z_3\overline{z_1} =z_1+z_1+z_3^2=2z_1+z_3^2.\\[6pt] &\text{Since\; } z_1=\tfrac{1}{\sqrt2}-\tfrac{i}{\sqrt2},\quad z_3^2=e^{i\pi/2}=i,\\[4pt] &\Rightarrow S=2\Big(\tfrac{1}{\sqrt2}-\tfrac{i}{\sqrt2}\Big)+i=\sqrt2+i(1-\sqrt2).\\[6pt] &\therefore |S|^2=(\sqrt2)^2+(1-\sqrt2)^2=2+(1-2\sqrt2+2)=5-2\sqrt2. \end{aligned}

Thus $|S|^2=\alpha+\beta\sqrt2$ with $\alpha=5,\ \beta=-2$ , and

\alpha^2+\beta^2=5^2+(-2)^2=25+4=29.

\boxed{29}

Answer key and solution verified before publishing.

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Exam: JEE Main 2025
Paper: 22 January, Morning Shift
Subject: Mathematics
Chapter: Complex Numbers
Topic: Geometry of Complex Numbers