JEE MAIN - Mathematics (2024 - 6th April Evening Shift - No. 2)
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
either $$z_1$$ lies on a circle of radius $$\frac{1}{2}$$ or $$z_2$$ lies on a circle of radius 1.
$$z_1$$ lies on a circle of radius $$\frac{1}{2}$$ and $$z_2$$ lies on a circle of radius 1.
either $$z_1$$ lies on a circle of radius 1 or $$z_2$$ lies on a circle of radius $$\frac{1}{2}$$.
both $$z_1$$ and $$z_2$$ lie on the same circle.
Explanation
$$\begin{aligned}
& \left|\frac{z_1-2 z_2}{\frac{1}{2}-z_1 \bar{z}_2}\right|=2 \\
& \left|z_1-2 z_2\right|=\left|1-2 z_1 \bar{z}_2\right| \\
& \Rightarrow\left(z_1-2 z_2\right)\left(\bar{z}_1-2 \bar{z}_2\right)=\left(1-2 z_1 \bar{z}_2\right)\left(1-2 \bar{z}_1 z_2\right) \\
& \Rightarrow\left|z_1\right|^2+4\left|z_2\right|^2-2 \bar{z}_1 z_2-2 \bar{z}_2 z_1 \\
& \quad=1+4\left|z_1\right|^2\left|z_2\right|^2-2 z_1 \bar{z}_2-2 \bar{z}_1 z_2 \\
& \Rightarrow\left|z_1\right|^2+4\left|z_2\right|^2-4\left|z_1\right|^2\left|z_2\right|^2-1=0 \\
& \Rightarrow\left(\left|z_1\right|^2-1\right)\left(1-4\left|z_2\right|^2\right)=0 \\
& \Rightarrow\left|z_1\right|=1 \text { and }\left|z_2\right|=\frac{1}{2}
\end{aligned}$$
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