Consider the hyperbola $\mathrm{H}: \mathrm{x}^2-\mathrm{y}^2=1$ and a circle S with center $\mathrm{N}\left(\mathrm{x}_2, 0\right)$. Suppose that H and S touch each other at a point $\mathrm{P}\left(\mathrm{x}_1, \mathrm{y}_1\right)$ with $\mathrm{x}_1>1$ and $\mathrm{y}_1>0$. The common tangent to H and S at P intersects the x -axis at point M . If $(1, m)$ is the centroid of the triangle $\Delta \mathrm{PMN}$, then the correct expression(s) is(are)
Select ALL correct options:
A
$\frac{\mathrm{d} l}{\mathrm{dx}_1}=1-\frac{1}{3 \mathrm{x}_1^2}$ for $\mathrm{x}_1>1$
B
$\frac{\mathrm{d} m}{\mathrm{dx}_1}=\frac{\mathrm{x}_1}{3\left(\sqrt{\mathrm{x}_1^2-1}\right)}$ for $\mathrm{x}_1>1$
C
$\frac{\mathrm{d} l}{\mathrm{dx}_1}=1+\frac{1}{3 \mathrm{x}_1^2}$ for $\mathrm{x}_1>1$
D
$\frac{\mathrm{d} m}{\mathrm{dy}_1}=\frac{1}{3}$ for $\mathrm{y}_1>0$
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