Question

If \( Z_1, Z_2, Z_3 \) are three complex numbers with unit modulus such that \[ |Z_1 - Z_2|^2 + |Z_1 - Z_3|^2 = 4 \] then \[ Z_1 Z_2 + \overline{Z_1} Z_2 + Z_1 Z_3 + \overline{Z_1} Z_3 = \]

• A. \( 0 \)
• B. \( |Z_2|^2 + |Z_3|^2 \)
• C. \( |Z_2|^2 - |Z_2 + Z_3|^2 \)
• D. \( 1 \)

Hint:

For complex numbers with unit modulus, use modulus properties to simplify expressions. Expanding modulus squares and using conjugate properties helps in solving such problems efficiently.

Answer 1

The Correct Option is A

Step 1: Understanding the Given Condition 
We are given three complex numbers \( Z_1, Z_2, Z_3 \) with unit modulus, i.e., \[ |Z_1| = |Z_2| = |Z_3| = 1. \] Also, we are given the equation: \[ |Z_1 - Z_2|^2 + |Z_1 - Z_3|^2 = 4. \] 

Step 2: Expanding the Modulus Expressions 
Expanding the squared modulus terms: \[ (Z_1 - Z_2)(\overline{Z_1 - Z_2}) + (Z_1 - Z_3)(\overline{Z_1 - Z_3}) = 4. \] Using properties of conjugates, \[ |Z_1|^2 + |Z_2|^2 - Z_1 \overline{Z_2} - \overline{Z_1} Z_2 + |Z_1|^2 + |Z_3|^2 - Z_1 \overline{Z_3} - \overline{Z_1} Z_3 = 4. \] Since \( |Z_1|^2 = |Z_2|^2 = |Z_3|^2 = 1 \), we simplify: \[ 1 + 1 - Z_1 \overline{Z_2} - \overline{Z_1} Z_2 + 1 + 1 - Z_1 \overline{Z_3} - \overline{Z_1} Z_3 = 4. \] 

Step 3: Evaluating the Summation 
Rearranging terms: \[ 4 - (Z_1 \overline{Z_2} + \overline{Z_1} Z_2 + Z_1 \overline{Z_3} + \overline{Z_1} Z_3) = 4. \] Thus, \[ Z_1 \overline{Z_2} + \overline{Z_1} Z_2 + Z_1 \overline{Z_3} + \overline{Z_1} Z_3 = 0. \]