If `z_1ne-z_2` and `|z_1+z_2|=|1/z_1 + 1/z_2|` then :

Options

A. at least one of `z_1,z_2` is unimodular
B. both `z_1,z_2` are unimodular
C. `z_1 . z_2` is unimodular
D. `z_1 - z_2` is unimodular

Answer

To solve the problem, we are given that \( z_1 \neq -z_2 \) and \( |z_1 + z_2| = \left| \frac{1}{z_1} + \frac{1}{z_2} \right| \). We need to analyze this condition step by step.

### Step 1: Rewrite the given condition
We start with the equation:
\[ |z_1 + z_2| = \left| \frac{1}{z_1} + \frac{1}{z_2} \right| \]
We can rewrite the right-hand side:
\[ \frac{1}{z_1} + \frac{1}{z_2} = \frac{z_2 + z_1}{z_1 z_2} \]
Thus, we have:
\[ |z_1 + z_2| = \left| \frac{z_1 + z_2}{z_1 z_2} \right| \]

### Step 2: Simplify the equation
Since \( z_1 + z_2 \neq 0 \) (because \( z_1 \neq -z_2 \)), we can divide both sides by \( |z_1 + z_2| \):
\[ 1 = \frac{1}{|z_1 z_2|} \]
This implies:
\[ |z_1 z_2| = 1 \]

### Step 3: Conclusion about unimodularity
The condition \( |z_1 z_2| = 1 \) indicates that the product \( z_1 z_2 \) is unimodular. Therefore, we conclude that both \( z_1 \) and \( z_2 \) must also be unimodular (i.e., their moduli are 1).

### Final Result
Thus, the answer to the question is that both \( z_1 \) and \( z_2 \) are unimodular.