If ` z_(1) = 1 +iand z_(2) = -3+2i` then ` lm ((z_(1)z_(2))/barz_(1))` is

Options

A. 2
B. `-3`
C. 3
D. `-2`

Answer

To solve the problem, we need to find the imaginary part of the expression \(\frac{z_1 z_2}{\bar{z_1}}\) where \(z_1 = 1 + i\) and \(z_2 = -3 + 2i\).

### Step-by-Step Solution:

1. **Identify the complex numbers**:
\[ z_1 = 1 + i, \quad z_2 = -3 + 2i \]

2. **Calculate the conjugate of \(z_1\)**:
\[ \bar{z_1} = 1 - i \]

3. **Multiply \(z_1\) and \(z_2\)**:
\[ z_1 z_2 = (1 + i)(-3 + 2i) \]
Using the distributive property:
\[ = 1 \cdot (-3) + 1 \cdot (2i) + i \cdot (-3) + i \cdot (2i \]
\[ = -3 + 2i - 3i + 2i^2 \]
Since \(i^2 = -1\):
\[ = -3 + 2i - 3i - 2 = -5 - i \]

4. **Now substitute back into the expression**:
\[ \frac{z_1 z_2}{\bar{z_1}} = \frac{-5 - i}{1 - i} \]

5. **Rationalize the denominator**:
Multiply the numerator and denominator by the conjugate of the denominator:
\[ = \frac{(-5 - i)(1 + i)}{(1 - i)(1 + i)} \]
Calculate the denominator:
\[ (1 - i)(1 + i) = 1^2 - i^2 = 1 - (-1) = 2 \]
Calculate the numerator:
\[ (-5 - i)(1 + i) = -5 \cdot 1 + (-5) \cdot i - i \cdot 1 - i \cdot i \]
\[ = -5 - 5i - i + 1 = -4 - 6i \]

6. **Combine the results**:
\[ = \frac{-4 - 6i}{2} = -2 - 3i \]

7. **Identify the imaginary part**:
The imaginary part of \(-2 - 3i\) is \(-3\).

### Final Answer:
The imaginary part of \(\frac{z_1 z_2}{\bar{z_1}}\) is \(-3\).