Complex Numbers
Real Part of Möbius Map on Unit Circle
Complex Numbers_PYQ
Grade 11
Question:
If $|z|=1$ and $w=\dfrac{z-1}{z+1}$ (where $z\neq-1$), then $\text{Re}(w)$ is
$0$
$\dfrac{1}{|z+1|^2}$
$\dfrac{1}{|z+1|}\cdot\dfrac{1}{|z+1|^2}$
$\dfrac{\sqrt{2}}{|z+1|^2}$
Step-by-Step Solution
Key Concept: For $|z|=1$, the Möbius map $w=(z-1)/(z+1)$ satisfies $\bar{w}=-w$ (i.e., $w$ is purely imaginary), which follows instantly from $\bar{z}=1/z$.
**Step 1: Use $|z|=1 \Rightarrow \bar{z}=1/z$**
Take the conjugate of $w$: $\bar{w}=\dfrac{\bar{z}-1}{\bar{z}+1}=\dfrac{1/z-1}{1/z+1}=\dfrac{1-z}{1+z}=-w$.
**Step 2: Conclude**
$w+\bar{w}=0 \Rightarrow 2\,\text{Re}(w)=0 \Rightarrow \text{Re}(w)=0$.
Correct Answer: 1