Consider the hyperbola \( H: x^{2}-y^{2}=1 \) and a circle \( S \) with centre \( N\left(x_{2}, 0\right) \). 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 \( P M N \), 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) \( \frac{d m}{d x_{1}}=\frac{x_{1}}{3\left(\sqrt{\left(x_{1}^{2}-1\right)}\right)} \) 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 \( y_{1}0 \)
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