Question

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

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

Hint:

Remember that \(\operatorname{Arg}(z^n)=n\operatorname{Arg}(z)\) (mod \(2\pi\)). Converting one complex variable in terms of another often simplifies argument problems.

Answer 1

The Correct Option is A

Step 1: Use the given relation between \(z\) and \(w\).
Given, \[ \overline{z}+i\overline{w}=0 \] Therefore, \[ \overline{z}=-i\overline{w} \] Taking conjugates on both sides, \[ z=iw \] Hence, \[ w=-iz \]

Step 2: Express \(zw\) in terms of \(z\).
Substituting \(w=-iz\), \[ zw=z(-iz) \] \[ =-iz^2 \] Taking arguments, \[ \operatorname{Arg}(zw) = \operatorname{Arg}(-i)+\operatorname{Arg}(z^2) \] \[ =-\frac{\pi}{2}+2\operatorname{Arg}(z) \]

Step 3: Use the condition \(\operatorname{Arg}(zw)=\pi\).
Given, \[ -\frac{\pi}{2}+2\operatorname{Arg}(z)=\pi \] Hence, \[ 2\operatorname{Arg}(z)=\frac{3\pi}{2} \] Therefore, \[ \operatorname{Arg}(z)=\frac{3\pi}{4} \]

Step 4: Final conclusion.
Thus, \[ \boxed{\operatorname{Arg}(z)=\frac{3\pi}{4}} \]