Let $z$ and $w$ be two non-zero complex numbers such that $|z| = |w|$ and $\arg(z) + \arg(w) = \pi$. Then $z$ equals
Step-by-Step Solution
Key Concept: When $|z|=|w|$ and $\arg(z)+\arg(w)=\pi$, the numbers are reflections of each other across the imaginary axis: $w=-\bar{z}$, equivalently $z=-\bar{w}$.
**Step 1: Write in polar form**
Let $z=re^{i\alpha}$ and $w=re^{i\beta}$ (same modulus $r$). Given $\alpha+\beta=\pi$, so $\beta=\pi-\alpha$.
**Step 2: Express w in terms of z**
$w = re^{i(\pi-\alpha)} = -re^{-i\alpha} = -\bar{z}$.
**Step 3: Solve for z**
$w=-\bar{z} \Rightarrow \bar{w}=-z \Rightarrow z=-\bar{w}$.
Correct Answer: 4