Question

If \( z_1 \) and \( z_2 \) are two complex numbers with \( |z_1| = 1 \), then \( \left| \frac{z_1 - z_2}{1 - z_1 \overline{z_2}} \right| \) is equal to:

• A. 0
• B. \( \frac{1}{4} \)
• C. \( \frac{1}{2} \)
• D. 1
• E. 2

Hint:

This result can be understood through the lens of complex transformation and geometric interpretations in the complex plane, where distances and angles are preserved under certain conditions.

Answer 1

The Correct Option is D

Given \( |z_1| = 1 \), we know from the properties of complex numbers that \( z_1 \overline{z_1} = 1 \). The expression in question can be rewritten using the properties of the modulus of complex numbers: \[ \left| \frac{z_1 - z_2}{1 - z_1 \overline{z_2}} \right| \] Using the property of modulus, the value of this expression can be further explored by recognizing it as a special case of the formula for the modulus of a Möbius transformation, where \( z_1 \) is on the unit circle. 
This formula holds true under the condition that the denominator does not become zero, which is when \( 1 - z_1 \overline{z_2} \neq 0 \). 
If \( z_1 \) is a unit complex number, the expression simplifies due to the rotational symmetry of complex numbers on the unit circle, maintaining the modulus value: \[ \left| \frac{z_1 - z_2}{1 - \overline{z_1} z_2} \right| = 1 \] 
This is because the transformation preserves the distance ratio due to its conformal nature, keeping the modulus invariant when \( |z_1| = 1 \).