Key Concepts and Formulas

• Polar Form of a Complex Number: A complex number $z$ can be represented as $z = re^{i\theta}$, where $r = |z|$ is the modulus and $\theta = \arg(z)$ is the argument.
• Properties of Conjugates: If $z = re^{i\theta}$, then its conjugate $\overline{z} = re^{-i\theta}$. Also, $\arg(\overline{z}) = -\arg(z)$.
• Properties of Modulus and Argument: For complex numbers $z_1$ and $z_2$, $|z_1z_2| = |z_1||z_2|$ and $\arg(z_1z_2) = \arg(z_1) + \arg(z_2)$.

Step-by-Step Solution

1. Express $z$ and $w$ in polar form based on the given conditions.

We are given that $|zw| = 1$ and $\arg(z) - \arg(w) = \frac{\pi}{2}$. Let $z = r_1e^{i\theta_1}$ and $w = r_2e^{i\theta_2}$. Then, $|z| = r_1$, $|w| = r_2$, $\arg(z) = \theta_1$, and $\arg(w) = \theta_2$.

• Using $|zw| = 1$: Since $|zw| = |z||w| = r_1r_2 = 1$, we have $r_1r_2 = 1$. This means $r_1 = \frac{1}{r_2}$ (or $r_2 = \frac{1}{r_1}$). Let's denote $r_1 = r$, then $r_2 = \frac{1}{r}$.
• Using $\arg(z) - \arg(w) = \frac{\pi}{2}$: We have $\theta_1 - \theta_2 = \frac{\pi}{2}$, which means $\theta_1 = \theta_2 + \frac{\pi}{2}$. Let's denote $\theta_2 = \theta$, then $\theta_1 = \theta + \frac{\pi}{2}$.

Therefore, we can write: $z = re^{i(\theta + \frac{\pi}{2})}$ $w = \frac{1}{r}e^{i\theta}$

2. Calculate $z\overline{w}$.

First, we find the conjugate of $w$: $\overline{w} = \frac{1}{r}e^{-i\theta}$

Now we calculate $z\overline{w}$: $z\overline{w} = re^{i(\theta + \frac{\pi}{2})} \cdot \frac{1}{r}e^{-i\theta} = e^{i(\theta + \frac{\pi}{2} - \theta)} = e^{i\frac{\pi}{2}}$

3. Simplify $e^{i\frac{\pi}{2}}$.

Using Euler's formula, $e^{ix} = \cos(x) + i\sin(x)$, we have: $e^{i\frac{\pi}{2}} = \cos\left(\frac{\pi}{2}\right) + i\sin\left(\frac{\pi}{2}\right) = 0 + i(1) = i$

Therefore, $z\overline{w} = i$.

4. Calculate $\overline{z}w$.

First, we find the conjugate of $z$: $\overline{z} = re^{-i(\theta + \frac{\pi}{2})}$

Now we calculate $\overline{z}w$: $\overline{z}w = re^{-i(\theta + \frac{\pi}{2})} \cdot \frac{1}{r}e^{i\theta} = e^{i(-\theta - \frac{\pi}{2} + \theta)} = e^{-i\frac{\pi}{2}}$

5. Simplify $e^{-i\frac{\pi}{2}}$.

Using Euler's formula, $e^{ix} = \cos(x) + i\sin(x)$, we have: $e^{-i\frac{\pi}{2}} = \cos\left(-\frac{\pi}{2}\right) + i\sin\left(-\frac{\pi}{2}\right) = 0 + i(-1) = -i$

Therefore, $\overline{z}w = -i$.

6. Check the Options

We found that $z\overline{w} = i$ and $\overline{z}w = -i$. Option (A) gives $z\overline w = \frac{1-i}{\sqrt{2}}$, which is incorrect. Option (B) gives $\overline z w = i$, which is incorrect. Option (C) gives $z\overline w = \frac{-1+i}{\sqrt{2}}$, which is incorrect. Option (D) gives $\overline z w = -i$, which is correct. However, the given "Correct Answer" is (A), which is incorrect. The correct answer should be (D).

7. Recalculate $z\overline{w}$ to match the given correct answer, assuming an error in the question.

Let's assume $\arg(z)-\arg(w) = \frac{3\pi}{4}$. Then $\theta_1 = \theta + \frac{3\pi}{4}$. $z = re^{i(\theta + \frac{3\pi}{4})}$ $w = \frac{1}{r}e^{i\theta}$ $\overline{w} = \frac{1}{r}e^{-i\theta}$ $z\overline{w} = re^{i(\theta + \frac{3\pi}{4})} \cdot \frac{1}{r}e^{-i\theta} = e^{i(\theta + \frac{3\pi}{4} - \theta)} = e^{i\frac{3\pi}{4}}$ $e^{i\frac{3\pi}{4}} = \cos(\frac{3\pi}{4}) + i\sin(\frac{3\pi}{4}) = -\frac{1}{\sqrt{2}} + i\frac{1}{\sqrt{2}} = \frac{-1+i}{\sqrt{2}}$ If the question intended $\arg(z) - \arg(w) = -\frac{\pi}{4}$, then $\theta_1 = \theta - \frac{\pi}{4}$. $z = re^{i(\theta - \frac{\pi}{4})}$ $w = \frac{1}{r}e^{i\theta}$ $\overline{w} = \frac{1}{r}e^{-i\theta}$ $z\overline{w} = re^{i(\theta - \frac{\pi}{4})} \cdot \frac{1}{r}e^{-i\theta} = e^{i(\theta - \frac{\pi}{4} - \theta)} = e^{-i\frac{\pi}{4}}$ $e^{-i\frac{\pi}{4}} = \cos(-\frac{\pi}{4}) + i\sin(-\frac{\pi}{4}) = \frac{1}{\sqrt{2}} - i\frac{1}{\sqrt{2}} = \frac{1-i}{\sqrt{2}}$ If the question intended $\arg(z) - \arg(w) = -\frac{\pi}{4}$, then option (A) would be correct.

Common Mistakes & Tips

• Sign Errors: Pay close attention to the signs when working with arguments and conjugates, especially when subtracting angles.
• Euler's Formula: Remember Euler's formula ($e^{ix} = \cos x + i \sin x$) to convert between exponential and rectangular forms of complex numbers.
• Modulus Condition: Don't forget to use the modulus condition $|zw| = 1$ to simplify the expressions for $|z|$ and $|w|$.

Summary

Given the conditions $|zw| = 1$ and $\arg(z) - \arg(w) = \frac{\pi}{2}$, we expressed $z$ and $w$ in polar form and calculated $z\overline{w}$ and $\overline{z}w$. We found that $z\overline{w} = i$ and $\overline{z}w = -i$. Based on this, option (D) is correct. However, if the question intended $\arg(z) - \arg(w) = -\frac{\pi}{4}$, then option (A) would be correct.

Final Answer

The final answer is \boxed{D}, which corresponds to option (D).