If |z1|≠1,z1−z21−z¯1z2=1, then

|z1−z2|2=|1−z¯1z2|2 ⇒ |z1|2+|z2|2−z1z¯2−z¯1z2=1+|z1¯|2 |z2|2−z¯1z2−z1z¯2⇒ |z1|2 |z2|2−|z1|2−|z2|2+1=0⇒ (|z1|2 − 1) (|z2|2−1)=0 As |z1| ≠ 1, we get |z2|2=1 or |z2| = 1

If |z1|≠1,z1−z21−z¯1z2=1, then

|z1−z2|2=|1−z¯1z2|2 ⇒ |z1|2+|z2|2−z1z¯2−z¯1z2=1+|z1¯|2 |z2|2−z¯1z2−z1z¯2⇒ |z1|2 |z2|2−|z1|2−|z2|2+1=0⇒ (|z1|2 − 1) (|z2|2−1)=0 As |z1| ≠ 1, we get |z2|2=1 or |z2| = 1

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If $| z_{1} | \neq 1, |\frac{z_{1} - z_{2}}{1 - {\bar{z}}_{1} z_{2}}| = 1,$ then

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$| z_{2} | = 1$

$| z_{2} | = 0$

$| z_{2} | > 1$

$0 < | z_{2} | < 1$

answer is A.

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Detailed Solution

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

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