If $z_1, z_2$ are two distinct complex number such that $\left|\frac{z_1-2 z_2}{\frac{1}{2}-z_1…

If $z_1, z_2$ are two distinct complex number such that $\left|\frac{z_1-2 z_2}{\frac{1}{2}-z_1 \bar{z}_2}\right|=2$, then
  1. $z_1$ lies on a circle of radius $\frac{1}{2}$ and $z_2$ lies on a circle of radius 1 .
  2. both $z_1$ and $z_2$ lie on the same circle. both $z_1$ and $z_2$ lie on the same circle.
  3. either $z_1$ lies on a circle of radius $\frac{1}{2}$ or $z_2$ lies on a circle of radius 1 .
  4. either $z_1$ lies on a circle of radius 1 or $z_2$ lies on a circle of radius $\frac{1}{2}$.

Solution

$\begin{aligned} & \frac{z_1-2 z_2}{\frac{1}{2}-z_1 \bar{z}_2} \times \frac{\bar{z}_1-2 \bar{z}_2}{\frac{1}{2}-\bar{z}_1 z_2}=4 \\ & \left|z_1\right|^2 2 z_1 \bar{z}_2-2 \bar{z}_1 z_2+4\left|z_2\right|^2 \\ & =4\left(\frac{1}{4}-\frac{\bar{z}_1 z_2}{2}-\frac{z_1 \bar{z}_2}{2}+\left|z_1\right|^2\left|z_2\right|^2\right)\end{aligned}$
$\begin{aligned} & \left(z, \bar{z}_1-1\right)\left(1-2 z_2 \cdot 2 \bar{z}_2\right)=0 \\ & \left(\left|z_1\right|^2-1\right)\left(\left|2 z_2\right|^2-1\right)=0\end{aligned}$

Asked in: JEE Main 2024 (06 Apr Shift 2)

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