Let $z$ and $w$ be non-zero complex numbers such that $zw = |z^2|$ and $|w| = \left|\frac{z}{w}\right| = 4$. If $w$ varies, then the perimeter of the locus of $z$ is
Step-by-Step Solution
Key Concept: The argument of a quotient equals the difference of arguments: $\arg(z_1/z_2) = \arg(z_1) - \arg(z_2)$.
Given $\arg\left(\frac{4}{z}\right)$ where $z = 5\left(-1\right)^{2/5}$. First, $\left(-1\right)^{2/5} = e^{i\pi \cdot 2/5}$, so $z = 5e^{i2\pi/5}$. Then $\frac{4}{z} = \frac{4}{5}e^{-i2\pi/5}$. The argument is $-\frac{2\pi}{5} = \frac{8\pi}{5}$ (converting to principal range), which equals $w = \frac{8\pi}{5}(-1)$ giving $w = -\frac{8\pi}{5}$. Therefore $5|w| = 5 \cdot \frac{8\pi}{5} = 8\pi$, so $5|w| = 5$.
Correct Answer: 5