If `z _(1) , z _(2) and z _(3)` are three complex numbers such that `|z _(1)| = | z _(2)| = | z _(3)| = |(1)/(z _(1)) + (1)/(z _(2)) + (1)/( z _(2))| = 1,` then `| z _(1) + z _(2) + z _(3)|` is

Options

A. equal to 1
B. equal than 1
C. greater than 3
D. equal to 3

Answer

To solve the problem, we need to analyze the given conditions for the complex numbers \( z_1, z_2, z_3 \).

### Step 1: Understand the given conditions
We know that:
1. \( |z_1| = |z_2| = |z_3| = 1 \)
2. \( \left| \frac{1}{z_1} + \frac{1}{z_2} + \frac{1}{z_3} \right| = 1 \)

Since \( |z_i| = 1 \) for \( i = 1, 2, 3 \), we can express the reciprocals as:
\[ \frac{1}{z_i} = \overline{z_i} \]
Thus, we can rewrite the second condition as:
\[ |\overline{z_1} + \overline{z_2} + \overline{z_3}| = 1 \]

### Step 2: Use properties of conjugates
Using the property of complex conjugates, we have:
\[ |\overline{z_1} + \overline{z_2} + \overline{z_3}| = |z_1 + z_2 + z_3| \]
This means:
\[ |z_1 + z_2 + z_3| = 1 \]

### Step 3: Conclusion
From the above analysis, we find that:
\[ |z_1 + z_2 + z_3| = 1 \]

Thus, the final answer is:
\[ |z_1 + z_2 + z_3| = 1 \]