If `z_1 + a_1 + ib_1 and z_2 = a_2 + ib_2` are complex such that `|z_1| = 1, |z_2|=2 and "Re" (z_1 z_2)=0`, then the pair of complex numbers `omega_(1) = a_(1) = (ia_2)/(2) and omega_(2) = 2b_(1) + ib_(2)` satisfy.

Options

A. `|omega_(1)|=1`
B. `|omega_(2)|=2`
C. Re `(omega_(1) omega_(2) )=0`
D. lm `(omega_(1) omega_(2) )=2`

Answer

To solve the problem, we need to analyze the given complex numbers and their properties step by step.

### Given:
1. \( z_1 = a_1 + ib_1 \) where \( |z_1| = 1 \)
2. \( z_2 = a_2 + ib_2 \) where \( |z_2| = 2 \)
3. \( \text{Re}(z_1 z_2) = 0 \)

### Step 1: Understanding the Moduli
Since \( |z_1| = 1 \), we have:
\[ |z_1| = \sqrt{a_1^2 + b_1^2} = 1 \implies a_1^2 + b_1^2 = 1 \]

Since \( |z_2| = 2 \), we have:
\[ |z_2| = \sqrt{a_2^2 + b_2^2} = 2 \implies a_2^2 + b_2^2 = 4 \]

### Step 2: Condition on Real Part
The condition \( \text{Re}(z_1 z_2) = 0 \) means that the real part of the product \( z_1 z_2 \) is zero. We can express this product:
\[ z_1 z_2 = (a_1 + ib_1)(a_2 + ib_2) = a_1 a_2 + i(a_1 b_2 + b_1 a_2) - b_1 b_2 \]
The real part is:
\[ \text{Re}(z_1 z_2) = a_1 a_2 - b_1 b_2 = 0 \implies a_1 a_2 = b_1 b_2 \]

### Step 3: Finding \( \omega_1 \) and \( \omega_2 \)
Given:
\[ \omega_1 = \frac{ia_2}{2}, \quad \omega_2 = 2b_1 + ib_2 \]

### Step 4: Analyzing \( \omega_1 \)
From \( \omega_1 = \frac{ia_2}{2} \):
- The real part of \( \omega_1 \) is \( 0 \).
- The imaginary part is \( \frac{a_2}{2} \).

### Step 5: Analyzing \( \omega_2 \)
From \( \omega_2 = 2b_1 + ib_2 \):
- The real part is \( 2b_1 \).
- The imaginary part is \( b_2 \).

### Step 6: Conditions on \( \omega_1 \) and \( \omega_2 \)
We need to check if \( \omega_1 \) and \( \omega_2 \) satisfy any specific conditions.

1. **Modulus of \( \omega_1 \)**:
\[ |\omega_1| = \left| \frac{ia_2}{2} \right| = \frac{|a_2|}{2} \]

2. **Modulus of \( \omega_2 \)**:
\[ |\omega_2| = \sqrt{(2b_1)^2 + b_2^2} = \sqrt{4b_1^2 + b_2^2} \]

### Step 7: Conclusion
We have established the relationships and conditions for \( \omega_1 \) and \( \omega_2 \). The conditions derived from the properties of \( z_1 \) and \( z_2 \) lead us to understand how these complex numbers interact.