Consider the hyperbola $H : x^{2} - y^{2} = 1$ and a circle S with center $N (x_{2}, 0)$ . 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 $Δ P M N$ , then the correct expression(s) is(are)

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$\frac{d l}{d x_{1}} = 1 + \frac{1}{3 {x_{1}}^{2}} f o r x_{1} > 1$

$\frac{d m}{d x_{1}} = \frac{x_{1}}{3 (\sqrt{{x_{1}}^{2} - 1})} f o r x_{1} > 1$

$\frac{d m}{d y_{1}} = \frac{1}{3} f o r y_{1} > 0$

$\frac{d l}{d x_{1}} = 1 - \frac{1}{3 {x_{1}}^{2}} f o r x_{1} > 1$

answer is A, B, D.

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Detailed Solution

Equation tangent to H at P is $x x_{1} - y y_{1} = 1$

$l = \frac{x_{1} + x_{2} + \frac{1}{x_{1}}}{3}, m = \frac{y_{1}}{3} = \frac{\sqrt{{x_{1}}^{2} - 1}}{3}$

Now, $\frac{d y}{d x} {|_{H a t P} = \frac{d y}{d x}|}_{S a t P} \Rightarrow \frac{x_{1}}{y_{1}} = \frac{x_{2} - x_{1}}{y_{1}} \Rightarrow x_{2} = 2 x_{1}$

So, $l = x_{1} + \frac{1}{3 x_{1}}$

$\frac{d l}{d x_{1}} = 1 - \frac{1}{3 {x_{1}}^{2}}, \frac{d m}{d y_{1}} = \frac{1}{3}; \frac{d m}{d x_{1}} = \frac{1}{3} . \frac{x_{1}}{\sqrt{{x_{1}}^{2} - 3}}$