If z 1 , z 2 are two distinct complex number such that | z 1 - 2 z 2 1 2 - z 1 z ¯ 2 | = 2 , then [2024]

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Solution

Q.
If $z_{1}, z_{2}$ are two distinct complex number such that $| \frac{z_{1} - 2 z_{2}}{\frac{1}{2} - z_{1} {\bar{z}}_{2}} |$ $= 2$ , then [2024]

1 $z_{1}$ lies on a circle of radius $\frac{1}{2}$ and $z_{2}$ lies on a circle of radius 1.

2 either $z_{1}$ lies on a circle of radius 1 or $z_{2}$ lies on a circle of radius $\frac{1}{2} .$

3 both $z_{1}$ and $z_{2}$ lie on the same circle.

4 either $z_{1}$ lies on a circle of radius $\frac{1}{2}$ or $z_{2}$ lies on a circle of radius 1.

Ans.
(2)

We have, $| \frac{z_{1} - 2 z_{2}}{\frac{1}{2} - z_{1} {\bar{z}}_{2}} |$ $= 2$

$\Rightarrow | z_{1} - 2 z_{2} | = | 1 - 2 z_{1} {\bar{z}}_{2} |$

$\Rightarrow | z_{1} - 2 z_{2} |^{2} = | 1 - 2 z_{1} {\bar{z}}_{2} |^{2}$

$\Rightarrow (z_{1} - 2 z_{2}) ({\bar{z}}_{1} - 2 {\bar{z}}_{2}) = (1 - 2 z_{1} {\bar{z}}_{2}) (1 - 2 {\bar{z}}_{1} z_{2})$

$\Rightarrow | z_{1} |^{2} + 4 | z_{2} |^{2} - 2 {\bar{z}}_{1} z_{2} - 2 {\bar{z}}_{2} z_{1}$

$= 1 + 4 | z_{1} |^{2} | z_{2} |^{2} - 2 z_{1} {\bar{z}}_{2} - 2 {\bar{z}}_{1} z_{2}$

$\Rightarrow | z_{1} |^{2} + 4 | z_{2} |^{2} - 4 | z_{1} |^{2} | z_{2} |^{2} - 1 = 0$

$\Rightarrow (| z_{1} |^{2} - 1) (1 - 4 | z_{2} |^{2}) = 0$

$\Rightarrow | z_{1} | = 1, | z_{2} | = \frac{1}{2}$

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