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If $|Z - \frac{1}{Z}| = 2$ then the least and greatest values of $|Z|$ are respectively

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$\sqrt{2} - 1, \sqrt{2} + 1$

$0, \sqrt{2} + 1$

none of these

$\sqrt{2} - 1, \sqrt{2}$

answer is A.

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Detailed Solution

$|z - \frac{1}{z}| = 2$

$\Rightarrow | z | = |z - \frac{1}{z} + \frac{1}{z}|$

$\Rightarrow | z | \leq |z - \frac{1}{z}| + \frac{1}{| z |}$

$\Rightarrow | z | \leq 2 + \frac{1}{| z |}$

$\Rightarrow | z |^{2} \leq 2 | z | + 1$

$∴ | z |^{2} - 2 | z | + 1 \leq 2$

$\Rightarrow (| z | - 1)^{2} \leq 2$

$\Rightarrow | z | - 1 \leq \sqrt{2}$

$∴ | z | \leq \sqrt{2} + 1$

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