Question

Let \( z \) and \( w \) be two complex numbers such that \( \bar{z} + i\bar{w} = 0 \) and \( \text{Arg}(zw) = \pi \). Then \( \text{Arg}(z) = \)

• A. \( \frac{3\pi}{4} \)
• B. \( \frac{\pi}{2} \)
• C. \( \frac{5\pi}{4} \)
• D. \( \frac{\pi}{4} \)

Hint:

Complex Argument Rules}
Use the identity \( \text{Arg}(ab) = \text{Arg}(a) + \text{Arg}(b) \).
The argument of \( i \) is \( \frac{\pi}{2} \).
Conjugation reverses the sign of the argument: \( \text{Arg}(\bar{z}) = -\text{Arg}(z) \).

Answer 1

The Correct Option is A

Step 1: We are given: \[ \bar{z} + i\bar{w} = 0 \quad \text{and} \quad \text{Arg}(zw) = \pi. \] Step 2: Take conjugate of both sides: \[ z - iw = 0 \Rightarrow z = iw. \] Step 3: Multiply both sides by \( w \): \[ zw = iw^2. \] Step 4: Using properties of argument: \[ \text{Arg}(zw) = \text{Arg}(i) + \text{Arg}(w^2) = \frac{\pi}{2} + 2\text{Arg}(w). \] Step 5: Given \( \text{Arg}(zw) = \pi \), we get: \[ \frac{\pi}{2} + 2\text{Arg}(w) = \pi \Rightarrow 2\text{Arg}(w) = \frac{\pi}{2} \Rightarrow \text{Arg}(w) = \frac{\pi}{4}. \] Step 6: Since \( z = iw \), then: \[ \text{Arg}(z) = \text{Arg}(i) + \text{Arg}(w) = \frac{\pi}{2} + \frac{\pi}{4} = \frac{3\pi}{4}. \]