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 $x_1>1\text{and}{y}_1>0$ 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 $\Delta PMN$ then the correct expression is (A) $\frac{dl}{{dx}_1}=1-\frac{1}{3x_1^2}$ for $x_1>1$ (B) $\frac{dm}{{dx}_1}=\frac{x_{!}}{3\left({\sqrt{x}}_1^2-1\right)})f\text{or}{x}_1>1$ (C) $\frac{dl}{{dx}_1}=1+\frac{1}{3x_1^2}f\text{or}{x}_1>1$ (D) $\frac{dm}{{dy}_1}=\frac{1}{3}f\text{or}{y}_1>0$

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