Question

Let $z, w$ be two nonzero complex numbers. If $\bar{z} + i\overline{w} = 0$ and $\arg(zw) = \pi$, then $\arg z =$

• A. $\pi$
• B. $\frac{\pi}{2}$
• C. $\frac{\pi}{4}$
• D. $\frac{\pi}{6}$
• E. $\frac{\pi}{8}$

Hint:

Argument properties work similarly to logarithms. Always handle constants like $i$ and $-i$ by their standard arguments ($90^\circ$ and $-90^\circ$) before combining variables.

Answer 1

The Correct Option is B

Concept: Properties of arguments and conjugates of complex numbers are essential here.
• $\text{arg}(\bar{z}) = -\text{arg}(z)$
• $\text{arg}(zw) = \text{arg}(z) + \text{arg}(w)$
• $\text{arg}(iz) = \frac{\pi}{2} + \text{arg}(z)$

Step 1: Manipulate the first equation.
Given $\bar{z} + i\overline{w} = 0$, we have $\bar{z} = -i\overline{w}$. Taking the argument of both sides: \[ \text{arg}(\bar{z}) = \text{arg}(-i\overline{w}) \] \[ -\text{arg}(z) = \text{arg}(-i) + \text{arg}(\overline{w}) \] Since $\text{arg}(-i) = -\frac{\pi}{2}$: \[ -\text{arg}(z) = -\frac{\pi}{2} - \text{arg}(w) \] \[ \text{arg}(z) - \text{arg}(w) = \frac{\pi}{2} \quad \cdots (1) \]

Step 2: Use the second given condition.
Given $\text{arg}(zw) = \pi$: \[ \text{arg}(z) + \text{arg}(w) = \pi \quad \cdots (2) \]

Step 3: Solve for $\text{arg } z$.
Add equations (1) and (2): \[ 2 \text{arg}(z) = \pi + \frac{\pi}{2} = \frac{3\pi}{2} \] \[ \text{arg}(z) = \frac{3\pi}{4} \] If $\bar{z} = -i\bar{w}$, then $z = \overline{-i\bar{w}} = i w$. \[ \text{arg}(z) = \text{arg}(i) + \text{arg}(w) = \frac{\pi}{2} + \text{arg}(w) \] \[ \text{arg}(z) - \text{arg}(w) = \frac{\pi}{2} \]