If z and w are complex numbers satisfying `barz+ibarw=0` and `amp(zw)=pi`, then `amp(w)` is equal to (where, `amp(w) in (-pi,pi]`)

Options

A. `(pi)/(4)`
B. `(-pi)/(4)`
C. `(pi)/(2)`
D. `(3pi)/(4)`

Answer

To solve the problem, we are given two complex numbers \( z \) and \( w \) that satisfy the equation \( \bar{z} + i \bar{w} = 0 \) and the condition \( \text{amp}(zw) = \pi \). We need to find the value of \( \text{amp}(w) \).

### Step-by-Step Solution:

1. **Start with the given equation**:
\[ \bar{z} + i \bar{w} = 0 \]

2. **Rearranging the equation**:
\[ \bar{z} = -i \bar{w} \]

3. **Taking the conjugate of both sides**:
\[ z = -i w \]

4. **Express \( z \) in terms of \( w \)**:
\[ z = i w \]

5. **Using the property of amplitude**:
The amplitude of the product of two complex numbers can be expressed as:
\[ \text{amp}(zw) = \text{amp}(z) + \text{amp}(w) \]

6. **Substituting \( z = i w \)**:
\[ \text{amp}(zw) = \text{amp}(i w) = \text{amp}(i) + \text{amp}(w) \]

7. **Finding the amplitude of \( i \)**:
The amplitude of \( i \) is \( \frac{\pi}{2} \). Therefore:
\[ \text{amp}(zw) = \frac{\pi}{2} + \text{amp}(w) \]

8. **Setting the equation equal to the given amplitude**:
Since we know that \( \text{amp}(zw) = \pi \):
\[ \frac{\pi}{2} + \text{amp}(w) = \pi \]

9. **Solving for \( \text{amp}(w) \)**:
\[ \text{amp}(w) = \pi - \frac{\pi}{2} = \frac{\pi}{2} \]

10. **Final adjustment**:
Since we need \( \text{amp}(w) \) in the range \( (-\pi, \pi] \), we can keep \( \frac{\pi}{2} \) as it is within this range.

### Conclusion:
Thus, the value of \( \text{amp}(w) \) is:
\[ \text{amp}(w) = \frac{\pi}{2} \]