Let `z,w` be complex numbers such that `barz+ibarw=0` and `arg zw=pi` Then `argz` equals

Options

A. `(pi)/(4)`
B. `(pi)/(2)`
C. `(3pi)/(4)`
D. `(5pi)/(4)`

Explanation

To solve the problem, we need to find the argument of the complex number \( z \) given the conditions \( \bar{z} + i \bar{w} = 0 \) and \( \arg(zw) = \pi \).

### Step-by-step Solution:

1. **Start with the first condition:**
\[ \bar{z} + i \bar{w} = 0 \]
Rearranging gives:
\[ \bar{z} = -i \bar{w} \]

2. **Taking the conjugate of both sides:**
\[ z = -i w \]
This implies that \( z \) can be expressed in terms of \( w \).

3. **Use the second condition:**
\[ \arg(zw) = \pi \]
This means that the product \( zw \) points in the opposite direction of the positive real axis.

4. **Express \( zw \) using the relation we found:**
\[ zw = z(-i z) = -iz^2 \]
Therefore, we can write:
\[ \arg(zw) = \arg(-iz^2) \]

5. **Calculate the argument:**
Using the property of arguments:
\[ \arg(-iz^2) = \arg(-i) + \arg(z^2) \]
The argument of \(-i\) is \(-\frac{\pi}{2}\) (or equivalently \(\frac{3\pi}{2}\)) and the argument of \(z^2\) is \(2\arg(z)\):
\[ \arg(zw) = -\frac{\pi}{2} + 2\arg(z) \]

6. **Set the equation equal to \(\pi\):**
\[ -\frac{\pi}{2} + 2\arg(z) = \pi \]

7. **Solve for \(\arg(z)\):**
\[ 2\arg(z) = \pi + \frac{\pi}{2} = \frac{3\pi}{2} \]
Dividing by 2 gives:
\[ \arg(z) = \frac{3\pi}{4} \]

### Final Answer:
\[ \arg(z) = \frac{3\pi}{4} \]