Let `z_(1) and z_(2)` be two non-zero complex number such that `|z_(1)|=|z_(2)|=|(1)/(z_(1)+(1)/(z_(2)))|=2`
What is the value of `|z_(1)+z_(2)|` ?

Options

A. 8
B. 4
C. 2
D. 1

Explanation

To solve the problem, we start with the given information about the complex numbers \( z_1 \) and \( z_2 \):

1. We know that \( |z_1| = |z_2| = 2 \).
2. We also have \( \left| \frac{1}{z_1} + \frac{1}{z_2} \right| = 2 \).

We need to find the value of \( |z_1 + z_2| \).

### Step 1: Express \( \frac{1}{z_1} + \frac{1}{z_2} \)

Using the property of complex numbers, we can express \( \frac{1}{z_1} + \frac{1}{z_2} \) as follows:

\[ \frac{1}{z_1} + \frac{1}{z_2} = \frac{z_2 + z_1}{z_1 z_2} \]

### Step 2: Find the modulus of \( \frac{1}{z_1} + \frac{1}{z_2} \)

Taking the modulus of both sides, we have:

\[ \left| \frac{1}{z_1} + \frac{1}{z_2} \right| = \left| \frac{z_1 + z_2}{z_1 z_2} \right| = \frac{|z_1 + z_2|}{|z_1||z_2|} \]

### Step 3: Substitute the known values

Since \( |z_1| = |z_2| = 2 \), we can substitute these values into the equation:

\[ \left| \frac{1}{z_1} + \frac{1}{z_2} \right| = \frac{|z_1 + z_2|}{|z_1| \cdot |z_2|} = \frac{|z_1 + z_2|}{2 \cdot 2} = \frac{|z_1 + z_2|}{4} \]

### Step 4: Set up the equation

From the problem statement, we know that:

\[ \left| \frac{1}{z_1} + \frac{1}{z_2} \right| = 2 \]

Thus, we can set up the equation:

\[ \frac{|z_1 + z_2|}{4} = 2 \]

### Step 5: Solve for \( |z_1 + z_2| \)

To find \( |z_1 + z_2| \), we multiply both sides of the equation by 4:

\[ |z_1 + z_2| = 2 \cdot 4 = 8 \]

### Conclusion

Therefore, the value of \( |z_1 + z_2| \) is:

\[ \boxed{8} \]