If ` z_(1) z_(2)` are two complex numbers such that Im `( z_(1) + z_(2)) = 0 `, Im ` ( z_(1) z_(2))` = 0 then

Options

A. `z_(1) = - z_(2)`
B. ` z_(1) = z_(2)`
C. ` z_(1) = bar(z_(2))`
D. none of these

Answer

To solve the problem, we start by defining 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 imaginary part of \( z_1 + z_2 \) is given by:
\[ \text{Im}(z_1 + z_2) = \text{Im}((x_1 + i y_1) + (x_2 + i y_2)) = \text{Im}(x_1 + x_2 + i(y_1 + y_2)) = y_1 + y_2 \]
Since we are given that \( \text{Im}(z_1 + z_2) = 0 \), we have:
\[ y_1 + y_2 = 0 \]
This implies:
\[ y_2 = -y_1 \]

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

Next, 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 of \( z_1 z_2 \) is:
\[ \text{Im}(z_1 z_2) = x_1 y_2 + y_1 x_2 \]
Since we are given that \( \text{Im}(z_1 z_2) = 0 \), we have:
\[ x_1 y_2 + y_1 x_2 = 0 \]

### Step 3: Substitute \( y_2 = -y_1 \)

Substituting \( y_2 = -y_1 \) into the equation \( x_1 y_2 + y_1 x_2 = 0 \):
\[ x_1 (-y_1) + y_1 x_2 = 0 \]
This simplifies to:
\[ -y_1 x_1 + y_1 x_2 = 0 \]
Factoring out \( y_1 \):
\[ y_1 (x_2 - x_1) = 0 \]

### Step 4: Analyze the results

From \( y_1 (x_2 - x_1) = 0 \), we have two cases:

1. \( y_1 = 0 \): This means \( z_1 \) is a real number.
2. \( x_2 - x_1 = 0 \): This means \( x_2 = x_1 \), indicating that both complex numbers have the same real part.

### Conclusion

Thus, we conclude that the complex numbers \( z_1 \) and \( z_2 \) are either both real or they have the same real part with opposite imaginary parts.

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