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Question: If \|z<sub>1</sub>\| ¹ 1, \(\left| \frac{z_{1} - z_{2}}{1 - {\bar{z}}_{1}z_{2}} \right|\)= 1, then...

If |z1| ¹ 1, ∣z1−z21−zˉ1z2∣\left| \frac{z_{1} - z_{2}}{1 - {\bar{z}}_{1}z_{2}} \right|= 1, then

A

|z­2| = 1

B

|z2| = 0

C

|z2| > 1

D

0 < |z2| < 1

Answer

|z­2| = 1

Explanation

Solution

Sol. |z1 – z2|2 = ∣1−zˉ1z2∣2|1 - {\bar{z}}_{1}z_{2}|^{2}

Ž ∣z1∣2+∣zˉ1∣2|z_{1}|^{2} + |{\bar{z}}_{1}|^{2}– 2Re(z1zˉ2)(z_{1}{\bar{z}}_{2}) = 1 + ∣zˉ1∣2∣z2∣2|{\bar{z}}_{1}|^{2}|z_{2}|^{2} – 2Re (zˉ1z2)({\bar{z}}_{1}z_{2})

or |z­1|2 + |z­2|2 – 1 – |z1|2 |z­2|2 = 0