Let z, w be complex numbers such that $\overline{z}+i\overline{w}=0$ and $arg(zw)=\pi.$ Then $arg(z)$ equals:

Question

Accepted Answer

To solve the problem, we need to analyze the given conditions for the complex numbers $ z $ and $ w $.

1. **Condition 1:** $\overline{z} + i\overline{w} = 0$

This implies that $\overline{z} = -i\overline{w}$. Taking the conjugate of both sides, we get $ z = iw $.

2. **Condition 2:** $\arg(zw) = \pi$

This means that the product $ zw $ is a negative real number. Since $\arg(zw) = \pi$, the angle of the product is $\pi$.

Now, let's substitute $ z = iw $ into the product $ zw $:

$$ zw = (iw)w = iw^2 $$

Since $ \arg(zw) = \pi $, we have:

$$ \arg(iw^2) = \pi $$

The argument of $ iw^2 $ can be broken down as follows:

- The argument of $ i $ is $\frac{\pi}{2}$.
- Let the argument of $ w $ be $\theta$. Then the argument of $ w^2 $ is $2\theta$.

Thus, the argument of $ iw^2 $ is:

$$ \arg(iw^2) = \frac{\pi}{2} + 2\theta = \pi $$

Solving for $\theta$:

$$ \frac{\pi}{2} + 2\theta = \pi $$

$$ 2\theta = \pi - \frac{\pi}{2} $$

$$ 2\theta = \frac{\pi}{2} $$

$$ \theta = \frac{\pi}{4} $$

Since $ z = iw $, the argument of $ z $ is:

$$ \arg(z) = \arg(iw) = \arg(i) + \arg(w) = \frac{\pi}{2} + \frac{\pi}{4} = \frac{3\pi}{4} $$

Therefore, the argument of $ z $ is $\frac{3\pi}{4}$, which corresponds to Option 3.

**Explanation for Incorrect Options:**

- **Option 1: $\frac{\pi}{4}$**

This would imply that the argument of $ z $ is less than the argument of $ w $, which contradicts the condition that $ \arg(z) = \arg(i) + \arg(w) $.

- **Option 2: $\frac{\pi}{2}$**

This would imply that the argument of $ z $ is equal to the argument of $ i $, which is not possible since $ z = iw $ and $\arg(w) = \frac{\pi}{4}$.

- **Option 4: $\frac{5\pi}{4}$**

This would imply that the argument of $ z $ is greater than $\pi$, which contradicts the condition that $\arg(z) = \frac{\pi}{2} + \frac{\pi}{4}$.

Thus, Option 3 is the correct answer, as it satisfies all the given conditions.