If `z_1,z_2` are two complex numbers such that `Im(z_1+z_2)=0,Im(z_1z_2)=0`, then:

Options

A. `z_(1) = -z_(2)`
B. `z_(1) = z_(2)`
C. `z_(1) = overline(z)_(2) `
D. None of these

Answer

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

Let:
- \( z_1 = x_1 + i y_1 \)
- \( z_2 = x_2 + i y_2 \)

### Step 1: Analyze the condition \( \text{Im}(z_1 + z_2) = 0 \)

The sum of the complex numbers is:
\[ z_1 + z_2 = (x_1 + x_2) + i(y_1 + y_2) \]
The imaginary part is:
\[ \text{Im}(z_1 + z_2) = y_1 + y_2 \]
Given that \( \text{Im}(z_1 + z_2) = 0 \), we have:
\[ y_1 + y_2 = 0 \implies y_1 = -y_2 \]

**Hint for Step 1:** Use the property of the imaginary part of a complex number to derive relationships between the components.

### Step 2: Analyze the condition \( \text{Im}(z_1 z_2) = 0 \)

Now, we calculate the product \( z_1 z_2 \):
\[ z_1 z_2 = (x_1 + i y_1)(x_2 + i y_2) = x_1 x_2 + i(x_1 y_2 + y_1 x_2) - y_1 y_2 \]
The imaginary part is:
\[ \text{Im}(z_1 z_2) = x_1 y_2 + y_1 x_2 \]
Given that \( \text{Im}(z_1 z_2) = 0 \), we have:
\[ x_1 y_2 + y_1 x_2 = 0 \]

**Hint for Step 2:** Set the imaginary part of the product to zero to find another relationship between the components.

### Step 3: Substitute \( y_1 \) in the equation

From Step 1, we know \( y_1 = -y_2 \). Substitute this into the equation from Step 2:
\[ x_1 y_2 + (-y_2) x_2 = 0 \]
This simplifies to:
\[ x_1 y_2 - y_2 x_2 = 0 \]
Factoring out \( y_2 \) gives:
\[ y_2 (x_1 - x_2) = 0 \]

**Hint for Step 3:** Factor the equation to identify possible cases for the variables.

### Step 4: Analyze the cases

1. **Case 1:** \( y_2 = 0 \)
- If \( y_2 = 0 \), then \( y_1 = 0 \) (from Step 1). Thus, both \( z_1 \) and \( z_2 \) are real numbers.

2. **Case 2:** \( x_1 - x_2 = 0 \)
- This implies \( x_1 = x_2 \).

From the first case, if both are real, they can be equal or opposite. From the second case, if \( x_1 = x_2 \), then \( z_2 = z_1 \) or \( z_2 = \overline{z_1} \) based on the imaginary parts being negatives of each other.

### Conclusion

Thus, we conclude that:
\[ z_1 = \overline{z_2} \]
This means that \( z_1 \) and \( z_2 \) are complex conjugates of each other.

**Final Answer:** The correct option is \( z_1 = \overline{z_2} \).