Mathematics

Problem

Consider the hyperbola

H: x^2 - y^2 = 1

and a circle S with centre

N(x_2, 0)

. Suppose that H and S 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 PMN, then the correct expression(s) is(are) a.

\frac{dl}{dx_1} = 1 - \frac{1}{3x_1^2}

for

x_1 > 1

\frac{dm}{dx_1} = \frac{x_1}{3(\sqrt{x_1^2 - 1})}

for

x_1 > 1

\frac{dl}{dx_1} = 1 + \frac{1}{3x_1^2}

for

x_1 > 1

\frac{dm}{dy_1} = \frac{1}{3}

for

y_1 > 0