Consider the hyperbola $$H : x^2 - y^2 = 1$$ and a circle S with center $$N(x_2, 0).$$ Suppose that H andS touch each other at a point $$P(x_1, y_1)$$ 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 $$\triangle PMN$$, then the correct expression(s) is(are)

A

$$\frac{dl}{dx_1} = 1 - \frac{1}{3_1^2}$$ for $$x_1 > 1$$

B

$$\frac{dm}{dx_1} = \frac{x_1}{3\left(\sqrt {x_1^2 - 1}\right)}$$ for $$x_1 > 1$$

C

$$\frac{dl}{dx_1} = 1 + \frac{1}{3_1^2}$$ for $$x_1 > 1$$

D

$$\frac{dm}{dy_1} = \frac{1}{3}$$ for $$y_1 > 0$$