If $zw = |z|^2$ and $zw = z\bar{z}$, find $|z|$ given that $|z| = 4$
Step-by-Step Solution
Key Concept: Use the relation $zw = |z|^2$ to find $w$, then apply modulus constraints to find the components of $z$
Given $zw = |z|^2$ and $zw = z\bar{z}$, we have $|z|^2 = z\bar{z}$. Also, $w = \frac{|z|^2}{z} = \bar{z}$. Now, $|z - \bar{z}| = |w| = 4$ leads to $|2i \cdot \text{Im}(z)| = 4$, so $|\text{Im}(z)| = 2$. Using the constraint $|z| = 4$, we get $|z|^2 = \text{Re}(z)^2 + \text{Im}(z)^2 = 16$, giving $\text{Re}(z)^2 = 12$. Therefore, the answer is $2$.
Correct Answer: 2