Let the complex numbers and 1 lie on the circles | z - z 0 | 2 = 4 and | z - z 0 | 2 = 16 respectively, where z 0 = 1 + i . Then, the value of is ________. [2024]

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Solution

Q.
Let the complex numbers $α$ and $\frac{1}{\bar{α}}$ lie on the circles $| z - z_{0} |^{2} = 4$ and $| z - z_{0} |^{2} = 16$ respectively, where $z_{0} = 1 + i .$ Then, the value of $100 | α |^{2}$ is ________. [2024]

Ans.
(20)

Given $| z - z_{0} |^{2} = 4$ ...(i)

$| z - z_{0} |^{2} = 16$ ...(ii)

$α$ lies on (i), $∴$ $| α - z_{0} |^{2} = 4$

   $\Rightarrow (α - z_{0}) (\bar{α} - {\bar{z}}_{0}) = 4 \Rightarrow α \bar{α} - α {\bar{z}}_{0} - z_{0} \bar{α} + | z_{0} |^{2} = 4$

   $\Rightarrow | α |^{2} - α {\bar{z}}_{0} - z_{0} \bar{α} + 2 = 4 (∵ | z_{0} | = \sqrt{2})$

   $\Rightarrow | α |^{2} - a {\bar{z}}_{0} - z_{0} \bar{α} = 2$ ...(iii)

$\frac{1}{\bar{α}}$ lies on (ii)

$∴$    $| \frac{1}{\bar{α}} - z_{0} |^{2} = 16 \Rightarrow (\frac{1}{\bar{α}} - z_{0}) (\frac{1}{α} - {\bar{z}}_{0}) = 16$

$\Rightarrow (1 - \bar{α} z_{0}) (1 - α {\bar{z}}_{0}) = 16 α \bar{α}$

$\Rightarrow 1 - \bar{α} z_{0} - α {\bar{z}}_{0} + α \bar{α}$ $z_{0} {\bar{z}}_{0} = 16 | α |^{2}$

$\Rightarrow 1 - \bar{α} z_{0} - α {\bar{z}}_{0} = 14 | α |^{2}$ ...(iv)

From (iii) and (iv), we get

$\Rightarrow 15 | α |^{2} = 3 \Rightarrow | α |^{2} = \frac{3}{15} = \frac{1}{5}$

$∴$ $100 | α |^{2} = 100 \times \frac{1}{5} = 20$

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