Verified Solution • 2025–26 Session moderate

QJEEM24Ma86fd: JEE Main 2024 Mathematics - Class 11 JEEM24Ma86fd Solution: If are two distinct complex number such that , then

If z1,z2z_1, z_2 are two distinct complex number such that ∣z1−2z212−z1zˉ2∣=2\left|\frac{z_1-2 z_2}{\frac{1}{2}-z_1 \bar{z}_2}\right|=2 , then

Multiple Choice Options

  • A either z1z_1 lies on a circle of radius 12\frac{1}{2} or z2z_2 lies on a circle of radius 1.
  • B z1z_1 lies on a circle of radius 12\frac{1}{2} and z2z_2 lies on a circle of radius 1.
  • C either z1z_1 lies on a circle of radius 1 or z2z_2 lies on a circle of radius 12\frac{1}{2} .
  • D both z1z_1 and z2z_2 lie on the same circle.

Step-by-Step Verified Solution

amp;∣z1−2z212−z1zˉ2∣=2amp;∣z1−2z2∣=∣1−2z1zˉ2∣amp;⇒(z1−2z2)(zˉ1−2zˉ2)=(1−2z1zˉ2)(1−2zˉ1z2)amp;⇒∣z1∣2+4∣z2∣2−2zˉ1z2−2zˉ2z1amp;=1+4∣z1∣2∣z2∣2−2z1zˉ2−2zˉ1z2amp;⇒∣z1∣2+4∣z2∣2−4∣z1∣2∣z2∣2−1=0amp;⇒(∣z1∣2−1)(1−4∣z2∣2)=0amp;⇒∣z1∣=1 and ∣z2∣=12\begin{aligned}& \left|\frac{z_1-2 z_2}{\frac{1}{2}-z_1 \bar{z}_2}\right|=2 \\& \left|z_1-2 z_2\right|=\left|1-2 z_1 \bar{z}_2\right| \\& \Rightarrow\left(z_1-2 z_2\right)\left(\bar{z}_1-2 \bar{z}_2\right)=\left(1-2 z_1 \bar{z}_2\right)\left(1-2 \bar{z}_1 z_2\right) \\& \Rightarrow\left|z_1\right|^2+4\left|z_2\right|^2-2 \bar{z}_1 z_2-2 \bar{z}_2 z_1 \\& \quad=1+4\left|z_1\right|^2\left|z_2\right|^2-2 z_1 \bar{z}_2-2 \bar{z}_1 z_2 \\& \Rightarrow\left|z_1\right|^2+4\left|z_2\right|^2-4\left|z_1\right|^2\left|z_2\right|^2-1=0 \\& \Rightarrow\left(\left|z_1\right|^2-1\right)\left(1-4\left|z_2\right|^2\right)=0 \\& \Rightarrow\left|z_1\right|=1 \text { and }\left|z_2\right|=\frac{1}{2}\end{aligned}

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