Let $n \ge 2$, and consider the inversion $\Phi\colon \mathbb{R}^n \setminus \{\vec{0}\} \to \mathbb{R}^n \setminus \{\vec{0}\}$ given by $x \mapsto \frac{x}{|x|^2}$. For $a > 0$, define the half space $H_a^+ = \{(x_1, \dots, x_{n-1}, x_n) \in \mathbb{R}^n | x_n > a\}$. Prove that $\Phi(H_a^+) = B((0, \dots, 0, \frac{1}{2a}), \frac{1}{2a})$. That is, we want to show that $\Phi(H_a^+)$ equals the $n$-dimensonal ball of radius $\frac{1}{2a}$, centered at the point $(0, \dots, 0, \frac{1}{2a}) \in \mathbb{R}^n$.
This question has appeared on an old PDE qualifying exam. In the course, we covered several topics, including the wave and heat equations, harmonic functions, the Dirichlet problem, and eigenfunctions of the Laplacian. I'm wondering if there is some technique that's related to these concepts and can be used to solve this problem.
One observation I have made is that the set $\{(0, \dots,0, r) \in \mathbb{R}^n| r > a\} \subseteq H_a^+$ is mapped onto $\{(0, \dots, 0, \frac{1}{r}) \in \mathbb{R}^n| r> a\} \subseteq B((0, \dots, 0, \frac{1}{2a}), \frac{1}{2a})$. I know there is much more to be done, but this is the best I have so far! Hints or solutions are greatly appreciated.