Given hyperbola $C : x ^ { 2 } - y ^ { 2 } = m ( m > 0 )$, point $P _ { 1 } ( 5,4 )$ is on $C$, and $k$ is a constant with $0 < k < 1$. Through $P_n$ on the right branch of $C$, draw a line with slope $k$; this line intersects $C$ at another point $Q_n$ on the left branch. The reflection of $Q_n$ across the $y$-axis gives $P_{n+1}$ on the right branch. (1) Find the coordinates of $P_2$ when $k = \frac{1}{2}$. (2) Prove that $\{x_n - y_n\}$ is a geometric sequence. (3) Prove that $S_n = \sum_{i=1}^{n} (x_i y_{i+1} - y_i x_{i+1})$ is a constant independent of $n$.
(1) $x_2 = 3, y_2 = 0$. (2) Using Vieta's formulas, $x_{n+1} - y_{n+1} = \frac{1-k}{1+k}(x_n - y_n)$, so $\{x_n - y_n\}$ is a geometric sequence with common ratio $\frac{1-k}{1+k}$. (3) $S_n$ is a constant independent of $n$.
Given hyperbola $C : x ^ { 2 } - y ^ { 2 } = m ( m > 0 )$, point $P _ { 1 } ( 5,4 )$ is on $C$, and $k$ is a constant with $0 < k < 1$. Through $P_n$ on the right branch of $C$, draw a line with slope $k$; this line intersects $C$ at another point $Q_n$ on the left branch. The reflection of $Q_n$ across the $y$-axis gives $P_{n+1}$ on the right branch.\\
(1) Find the coordinates of $P_2$ when $k = \frac{1}{2}$.\\
(2) Prove that $\{x_n - y_n\}$ is a geometric sequence.\\
(3) Prove that $S_n = \sum_{i=1}^{n} (x_i y_{i+1} - y_i x_{i+1})$ is a constant independent of $n$.