Key Concepts and Formulas

• Modulus of a Complex Number: For a complex number $z = x + yi$, the modulus is given by $|z| = \sqrt{x^2 + y^2}$. This represents the distance from the origin to the point $(x, y)$ in the complex plane.
• Argument of a Complex Number: The argument of a complex number $z$, denoted by $\arg(z)$, is the angle between the positive real axis and the line segment connecting the origin to the point representing $z$ in the complex plane.
• Modulus of Product: For complex numbers $z_1$ and $z_2$, $|z_1 z_2| = |z_1| |z_2|$.
• Area of a Triangle in the Complex Plane: The area of a triangle with vertices at the origin, $z_1$, and $z_2$ is given by $\frac{1}{2} |z_1| |z_2| |\sin(\arg(z_1) - \arg(z_2))|$. If the angle between $z_1$ and $z_2$ at the origin is $\frac{\pi}{2}$, then the area simplifies to $\frac{1}{2} |z_1| |z_2|$.

Step-by-Step Solution

Step 1: Analyze the Complex Number w and find its modulus

• Why this step: We need to find the modulus of $w$ because it is a side of the triangle, and its length is required to calculate the area.
• Given $w = 1 - \sqrt{3}i$.
• The modulus of $w$ is calculated as follows: $|w| = |1 - \sqrt{3}i| = \sqrt{1^2 + (-\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2$
• Therefore, $|w| = 2$.

Step 2: Determine the Modulus of z

• Why this step: We need to find the modulus of $z$ because it is another side of the triangle, and its length is required to calculate the area. We are given $|zw| = 1$, and we know $|w|$ from the previous step.
• We are given $|zw| = 1$. Using the property of moduli, $|z||w| = 1$.
• Substituting the value of $|w| = 2$ into the equation, we get: $|z| \cdot 2 = 1$ $|z| = \frac{1}{2}$
• Therefore, $|z| = \frac{1}{2}$.

Step 3: Determine the Angle Between z and w at the Origin

• Why this step: We need to determine the angle between the complex numbers $z$ and $w$ at the origin to use the area formula. We are given $\arg(z) - \arg(w) = \frac{\pi}{2}$.
• We are given that $\arg(z) - \arg(w) = \frac{\pi}{2}$.
• This means the angle between the line segments from the origin to $z$ and from the origin to $w$ is $\frac{\pi}{2}$ radians, or 90 degrees. The triangle formed by the origin, $z$, and $w$ is a right-angled triangle.

Step 4: Calculate the Area of the Triangle

• Why this step: Now that we have the lengths of two sides and the angle between them, we can calculate the area of the triangle.
• The area of the triangle with vertices at the origin, $z$, and $w$ is given by: $Area = \frac{1}{2} |z| |w| |\sin(\arg(z) - \arg(w))|$
• Since $\arg(z) - \arg(w) = \frac{\pi}{2}$, we have $\sin(\frac{\pi}{2}) = 1$. Therefore, $Area = \frac{1}{2} |z| |w|$
• Substituting $|z| = \frac{1}{2}$ and $|w| = 2$, we get: $Area = \frac{1}{2} \cdot \frac{1}{2} \cdot 2 = \frac{1}{2}$

Common Mistakes & Tips

• Modulus Calculation: Ensure correct squaring and addition when finding the modulus.
• Modulus Property: Remember the modulus property: $|z_1 z_2| = |z_1| |z_2|$.
• Argument Interpretation: Understand that $\arg(z) - \arg(w)$ represents the angle between the line segments from the origin to $z$ and $w$.

Summary We first found the modulus of $w$. Then, using the given condition $|zw|=1$, we found the modulus of $z$. We were given that $\arg(z) - \arg(w) = \frac{\pi}{2}$, which means the triangle formed by the origin, $z$, and $w$ is a right-angled triangle. Finally, we calculated the area of the triangle using the formula $\frac{1}{2}|z||w|$, which gave us an area of $\frac{1}{2}$. This corresponds to option (D).

Final Answer The final answer is $\boxed{1/2}$, which corresponds to option (D).