Consider the hyperbola $H : x ^ { 2 } - y ^ { 2 } = 1$ and a circle $S$ with center $N \left( x _ { 2 } , 0 \right)$. Suppose that $H$ and $S$ touch each other at a point $P \left( x _ { 1 } , y _ { 1 } \right)$ with $x _ { 1 } > 1$ and $y _ { 1 } > 0$. The common tangent to $H$ and $S$ at $P$ intersects the $x$-axis at point $M$. If ( $l , m$ ) is the centroid of the triangle $\triangle P M N$, then the correct expression(s) is(are) (A) $\frac { d l } { d x _ { 1 } } = 1 - \frac { 1 } { 3 x _ { 1 } ^ { 2 } }$ for $x _ { 1 } > 1$ (B) $\frac { d m } { d x _ { 1 } } = \frac { x _ { 1 } } { 3 \left( \sqrt { x _ { 1 } ^ { 2 } - 1 } \right) }$ for $x _ { 1 } > 1$ (C) $\frac { d l } { d x _ { 1 } } = 1 + \frac { 1 } { 3 x _ { 1 } ^ { 2 } }$ for $x _ { 1 } > 1$ (D) $\frac { d m } { d y _ { 1 } } = \frac { 1 } { 3 }$ for $y _ { 1 } > 0$
Consider the hyperbola $H : x ^ { 2 } - y ^ { 2 } = 1$ and a circle $S$ with center $N \left( x _ { 2 } , 0 \right)$. Suppose that $H$ and $S$ touch each other at a point $P \left( x _ { 1 } , y _ { 1 } \right)$ with $x _ { 1 } > 1$ and $y _ { 1 } > 0$. The common tangent to $H$ and $S$ at $P$ intersects the $x$-axis at point $M$. If ( $l , m$ ) is the centroid of the triangle $\triangle P M N$, then the correct expression(s) is(are)\\
(A) $\frac { d l } { d x _ { 1 } } = 1 - \frac { 1 } { 3 x _ { 1 } ^ { 2 } }$ for $x _ { 1 } > 1$\\
(B) $\frac { d m } { d x _ { 1 } } = \frac { x _ { 1 } } { 3 \left( \sqrt { x _ { 1 } ^ { 2 } - 1 } \right) }$ for $x _ { 1 } > 1$\\
(C) $\frac { d l } { d x _ { 1 } } = 1 + \frac { 1 } { 3 x _ { 1 } ^ { 2 } }$ for $x _ { 1 } > 1$\\
(D) $\frac { d m } { d y _ { 1 } } = \frac { 1 } { 3 }$ for $y _ { 1 } > 0$