Let $z_1$ and $z_2$ be two complex numbers such that $|z_2|=1$ and $\dfrac{z_1+z_2}{z_1-z_2} = \dfrac{2z_2^2-z_2+1}{2}$. Find the value of $|z_1-1|+|z_1+1|$.
Step-by-Step Solution
Key Concept: Write $z_2=e^{i\theta}$. From the given equation, derive $z_1$ as a function of $\theta$. Show that $z_1=\cos\theta\cdot\frac{\cos\theta-1}{2-\cos\theta+1}$... Ultimately $z_1$ is purely real and lies in $[-1,1]$.
$z_1$ lies in $[-1,1]\subset\mathbb{R}$, so $|z_1-1|+|z_1+1|=\mathbf{2}$.
Correct Answer: 2