Let $A, B, C$ be the set of complex numbers defined as $A = \{z: |z + 2| - |z - 2|| = 2\}$, $B = \left\{z: \arg\left(\frac{z - 1}{z}\right) = \frac{\pi}{2}\right\}$ and $C = \{z: \arg(z - 1) = \pi\}$, then $n(A \cap B \cap C) = $ ___________.
Step-by-Step Solution
Key Concept: The three sets $A$, $B$, $C$ represent a hyperbola, a circle, and a ray respectively; determining if they have a common point requires verifying simultaneous membership in all three geometric loci.
Set $A$ is defined by $||z+2| - |z-2|| = 2$, which represents a hyperbola with foci at $\pm 2$. For points on this hyperbola, the absolute difference of distances from the two foci equals 2. Set $B$ is defined by $\arg\left(\frac{z-1}{z}\right) = \frac{\pi}{2}$, meaning $\frac{z-1}{z}$ is purely imaginary. This simplifies to $\frac{z-1}{z} = ki$ for real $k \neq 0$, which gives points on a circle. Set $C$ is defined by $\arg(z-1) = \pi$, meaning $z-1$ lies on the negative real axis, so $C = \{z : z = 1 + t$ where $t < 0\}$ (the ray from 1 extending left). For $A \cap B \cap C$ to be non-empty, we need a point simultaneously on the hyperbola, the circle, and the ray. Checking the geometric constraints: the ray $C$ lies on the real axis (left of 1), and analyzing whether this ray intersects both $A$ and $B$ reveals no common point satisfies all three conditions.
Correct Answer: 0