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JEE Main 20246 AprilEvening Shift
Exam
JEE Main
Year
2024
Date
6 April
Shift
Evening Shift
Subject
Mathematics
Chapter
Complex Number

JEE Main 2024 · 6 April · Evening Shift · Mathematics · Complex Number

If z_1, z_2 are two distinct complex number such that | z_1-2 z_2...

If z1,z2z_1, z_2 are two distinct complex number such that \left|\frac{z_1-2 z_2}{\frac{1}{2}-z_1 \bar{z}_2}\right|=2$, thenIf z_1, z_2 are two distinct complex number such that | z_1-2 z_2 (1)/(2)-z_1 z _2 |=2 , then

Options

  1. Az1z_1 lies on a circle of radius 12\frac{1}{2} and z2z_2 lies on a circle of radius 1 .
  2. Bboth z1z_1 and z2z_2 lie on the same circle. both z1z_1 and z2z_2 lie on the same circle.
  3. Ceither z1z_1 lies on a circle of radius 12\frac{1}{2} or z2z_2 lies on a circle of radius 1 .
  4. Deither z1z_1 lies on a circle of radius 1 or z2z_2 lies on a circle of radius 12\frac{1}{2}.

Correct answer

D. either z1z_1 lies on a circle of radius 1 or z2z_2 lies on a circle of radius 12\frac{1}{2}.

Step-by-step solution

z1−2z212−z1zˉ2×zˉ1−2zˉ212−zˉ1z2=4∣z1∣22z1zˉ2−2zˉ1z2+4∣z2∣2=4(14−zˉ1z22−z1zˉ22+∣z1∣2∣z2∣2)\begin{aligned} & \frac{z_1-2 z_2}{\frac{1}{2}-z_1 \bar{z}_2} \times \frac{\bar{z}_1-2 \bar{z}_2}{\frac{1}{2}-\bar{z}_1 z_2}=4 \\ & \left|z_1\right|^2 2 z_1 \bar{z}_2-2 \bar{z}_1 z_2+4\left|z_2\right|^2 \\ & =4\left(\frac{1}{4}-\frac{\bar{z}_1 z_2}{2}-\frac{z_1 \bar{z}_2}{2}+\left|z_1\right|^2\left|z_2\right|^2\right)\end{aligned}<imgsrc="https://cdn−question−pool.getmarks.app/pyq/jeemain/XdhdhRrK1fT55MuNvKItQzeNjzOhL2Gxb43FDC68br8.original.fullsize.png"><br/><img src="https://cdn-question-pool.Quantrex.app/pyq/jee_main/XdhdhRrK1fT55MuNvKItQzeNjzOhL2Gxb43FDC68br8.original.fullsize.png"><br />(z,zˉ1−1)(1−2z2⋅2zˉ2)=0(∣z1∣2−1)(∣2z2∣2−1)=0\begin{aligned} & \left(z, \bar{z}_1-1\right)\left(1-2 z_2 \cdot 2 \bar{z}_2\right)=0 \\ & \left(\left|z_1\right|^2-1\right)\left(\left|2 z_2\right|^2-1\right)=0\end{aligned}$
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