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 need to analyze the given conditions involving the complex numbers \( z \) and \( w \).

### Step 1: Understanding the given equation
The equation provided is:
\[ \bar{z} + i \bar{w} = 0 \]
From this, we can express \( \bar{w} \) in terms of \( \bar{z} \):
\[ i \bar{w} = -\bar{z} \implies \bar{w} = -i \bar{z} \]

### Step 2: Relating \( z \) and \( w \)
Taking the conjugate of both sides, we have:
\[ w = -i z \]

### Step 3: Finding the amplitude of \( zw \)
We are given that:
\[ \text{amp}(zw) = \pi \]
Using the property of the argument of products of complex numbers, we can write:
\[ \text{amp}(z) + \text{amp}(w) = \pi \]

### Step 4: Expressing \( z \) in terms of \( w \)
Substituting \( w \) from step 2 into the equation:
\[ \text{amp}(z) + \text{amp}(-i z) = \pi \]
The amplitude of \( -i z \) can be expressed as:
\[ \text{amp}(-i z) = \text{amp}(z) - \frac{\pi}{2} \]
Thus, we have:
\[ \text{amp}(z) + \left(\text{amp}(z) - \frac{\pi}{2}\right) = \pi \]

### Step 5: Simplifying the equation
Combining the terms gives us:
\[ 2 \text{amp}(z) - \frac{\pi}{2} = \pi \]
Adding \( \frac{\pi}{2} \) to both sides:
\[ 2 \text{amp}(z) = \frac{3\pi}{2} \]
Dividing by 2:
\[ \text{amp}(z) = \frac{3\pi}{4} \]

### Step 6: Finding \( \text{amp}(w) \)
Substituting \( \text{amp}(z) \) back into the equation for \( \text{amp}(w) \):
\[ \frac{3\pi}{4} + \text{amp}(w) = \pi \]
Solving for \( \text{amp}(w) \):
\[ \text{amp}(w) = \pi - \frac{3\pi}{4} = \frac{\pi}{4} \]

### Final Answer
Thus, the amplitude of \( w \) is:
\[ \text{amp}(w) = \frac{\pi}{4} \]