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

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

1. Re(\(z_1 z_2\)) = 0
2. Re(\(z_1 + z_2\))

Let's analyze these conditions:

1. Condition 1: Re(\(z_1 z_2\)) = 0
   • This implies the product of the two complex numbers \( z_1 \) and \( z_2 \) is purely imaginary.
2. Condition 2: Re(\(z_1 + z_2\))
   • This condition is incomplete, but let us assume it should imply zero or another significant meaning.

We are given four options regarding the imaginary parts of \( z_1 \) and \( z_2 \):

• A. Im(\(z_1\)) > 0 and Im(\(z_2\)) > 0
• B. Im(\(z_1\)) < 0 and Im(\(z_2\)) > 0
• C. Im(\(z_1\)) > 0 and Im(\(z_2\)) < 0
• D. Im(\(z_1\)) < 0 and Im(\(z_2\)) < 0

Let's evaluate which options can satisfy the given conditions.

1. Evaluating Option A:
   • If both imaginations are positive, \( z_1 z_2 \) would have a positive and a negative imaginary component, making the real part non-zero typically. Hence, this option cannot satisfy the condition Re(\(z_1 z_2\)) = 0.
2. Evaluating Option B:
   • For Im(\(z_1\)) < 0 and Im(\(z_2\)) > 0, the multiplying will result in Re(\(z_1 z_2\)) = 0 more likely due to opposite imaginary signs. This option satisfies the conditions.
3. Evaluating Option C:
   • If Im(\(z_1\)) > 0 and Im(\(z_2\)) < 0, again the real part of the product can be zero as the imaginations subtract off. So, this can satisfy the condition also.
4. Evaluating Option D:
   • Both imaginary parts being negative would not typically lead to a real part of zero due to the product being purely imaginary condition. Thus, this option is not feasible.

Conclusion: The correct options are B and C, as these possibilities satisfy the condition Re(\(z_1 z_2\)) = 0 where the product of their imaginary parts results in a zero real component.