Question

For two non-zero complex numbers z₁ and z₂, if Re(z₁z₂) = 0 and Re(z₁ + z₂), then which of the following are possible?
A. Im(z₁) > 0 and Im(z₂) > 0 |
B. Im(z₁) < 0 and Im(z₂) > 0 
C. Im(z₁) > 0 and Im(z₂) < 0 
D. Im(z₁) < 0 and Im(z₂) < 0 
Choose the correct answer from the options given below

• A. B and C
• B. B and D
• C. A and B
• D. A and C

Answer 1

The Correct Option is A

Step 1: Representing complex numbers. 

Let \( z_1 = x_1 + i y_1 \) and \( z_2 = x_2 + i y_2 \), where \( x_1, x_2 \) are the real parts and \( y_1, y_2 \) are the imaginary parts of \( z_1 \) and \( z_2 \), respectively.

Step 2: Using the given conditions.

We are given that:

• \( \text{Re}(z_1 z_2) = 0 \) and \( \text{Re}(z_1 + z_2) = 0. \)

For \( z_1 z_2 = (x_1 + i y_1)(x_2 + i y_2) \), the real part is:

\( \text{Re}(z_1 z_2) = x_1 x_2 - y_1 y_2. \)

Thus, we have:

\( x_1 x_2 - y_1 y_2 = 0 \Rightarrow x_1 x_2 = y_1 y_2. \, \cdots (1) \)

For \( z_1 + z_2 = (x_1 + i y_1) + (x_2 + i y_2) \), the real part is:

\( \text{Re}(z_1 + z_2) = x_1 + x_2. \)

Thus, we have:

\( x_1 + x_2 = 0 \Rightarrow x_1 = -x_2. \, \cdots (2) \)

Step 3: Analyzing the conditions.

From equation (2), we know that \( x_1 = -x_2 \), meaning the real parts of \( z_1 \) and \( z_2 \) are opposite in sign.

From equation (1), we have \( x_1 x_2 = y_1 y_2 \), which means that the product of the real parts is equal to the product of the imaginary parts. For this to hold, \( y_1 \) and \( y_2 \) must have opposite signs, because the real parts are of opposite signs.

Thus, we conclude that \( y_1 \) and \( y_2 \) are of opposite signs, which means that the imaginary parts of \( z_1 \) and \( z_2 \) must satisfy the conditions:

• \( \text{Im}(z_1) < 0 \) and \( \text{Im}(z_2) > 0 \) or
• \( \text{Im}(z_1) > 0 \) and \( \text{Im}(z_2) < 0 \).

Step 4: Conclusion.

The correct options are B and C, as they correspond to the valid cases for the imaginary parts of \( z_1 \) and \( z_2 \).