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Consider the hyperbola \( H: x^{2}-y^{2}=1 \) and a circle \( \mathrm{S} \) with center \( N\left(x_{2}, 0\right) \). Suppose that \( \mathrm{H} \) and \( \mathrm{S} \) 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) \( \frac{\mathrm{d} \ell}{\mathrm{dx}_{1}}=1-\frac{1}{3 \mathrm{x}_{1}^{2}} \) for \( x_{1}1 \)
(b) \( \frac{\mathrm{dm}}{\mathrm{dx}_{1}}=\frac{\mathrm{x}_{1}}{3\left(\sqrt{\mathrm{x}_{1}^{2}-1}\right)} \) for \( x_{1}1 \)
(c) \( \frac{\mathrm{d} \ell}{\mathrm{dx}_{1}}=1+\frac{1}{3 \mathrm{x}_{1}^{2}} \) for \( x_{1}1 \)
(d) \( \frac{\mathrm{dm}}{\mathrm{dy}}=\frac{1}{3} \) for \( y_{1}0 \)
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