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JEE Main 202523 JanuaryEvening Shift
Exam
JEE Main
Year
2025
Date
23 January
Shift
Evening Shift
Subject
Mathematics
Chapter
Complex Number

JEE Main 2025 · 23 January · Evening Shift · Mathematics · Complex Number

The number of complex numbers z , satisfying |z|=1 and | z z + z z...

The number of complex numbers zz, satisfying |z|=1 and \left|\frac{z}{\bar{z}}+\frac{\bar{z}}{z}\right|=1$, is :

Options

  1. A4
  2. B8
  3. C10
  4. D6

Correct answer

B. 8

Step-by-step solution

∣z∣=1∣zzˉ+zˉz∣=1⇒∣z2+(zˉ)2∣=1\begin{aligned} & |z|=1 \\ & \left|\frac{z}{\bar{z}}+\frac{\bar{z}}{z}\right|=1 \\ & \Rightarrow\left|z^2+(\bar{z})^2\right|=1 \end{aligned} Let z=x+iyz=x+i y⇒∣(x+iy)2+(x−iy)2∣=1⇒∣2x2−2y2∣=1⇒∣x2−y2∣=12⇒x2−y2=±12 and  x2+y2=1\begin{aligned} & \Rightarrow\left|(x+i y)^2+(x-i y)^2\right|=1 \\ & \Rightarrow\left|^{2} x^2-2 y^2\right|=1 \\ & \Rightarrow\left|x^2-y^2\right|=\frac{1}{2} \\ & \Rightarrow x^2-y^2=\frac{ \pm 1}{2} \\ & \text{\text{ and } } x^2+y^2=1 \end{aligned}$CaseI:Case I:x^2-y^2=\frac{1}{2}CaseII:Case II:x^2-y^2=-\frac{1}{2}$
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