Question

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 $P(x_1,y_1)\text{with}{x}_1>1\text{and}{y}_1>0.$ 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 $△PMN,$ then the correct expression(s) is (are)

• A. $\frac{dl}{d{x}_1}=1-\frac{1}{3x_1^2}\text{for}{x}_1>1$
• B. $\frac{dm}{d{x}_1}=\frac{x_1}{3(\sqrt{x_1^2-1})}\text{for}{x}_1>1$
• C. $\frac{dl}{d{x}_1}=1+\frac{1}{3x_1^2}\text{for}{x}_1>1$
• D. $\frac{dm}{d{y}_1}=\frac{1}{3}\text{for}{y}_1>1$

Answer 1

Answer: (A), (B), (D)

$\frac{dm}{d{y}_1}=\frac{1}{3}\text{for}{y}_1>1$

Equation of tangent at $P(x_1,y_1)$ on hyperbola is $x{x}_1-y{y}_1=1$
As $M,N$ lies on $x-$axis therefore $y$coordinate of point $M,N$ should be zero.
$x{x}_1=1\Rightarrow x=\frac{1}{x_1}\therefore \text{point}M:(\frac{1}{x_1},0)$
Now equation of normal at $P(x_1,y_1)$ on hyperbola is
$y-y_1=-\frac{y_1}{x_1}(x-x_1)$
$\text{Put}y=0\Rightarrow x=2x_1\Rightarrow x_2=2x_1\therefore \text{point}N:(2x_1,0)$
Centroid of triangle $△PMN,$
$l=\frac{x_1+\frac{1}{x_1}+2x_1}{3}=x_1+\frac{1}{3x_1}m=\frac{y_1+0+0}{3}=\frac{y_1}{3}\frac{dm}{d{y}_1}=\frac{1}{3}\text{for}{x}_1>0\frac{dl}{d{x}_1}=1-\frac{1}{3x_1^2}\text{for}{x}_1>1$
Also
$m=\frac{y_1}{3}=\frac{1}{3}\sqrt{x_1^2-1}\frac{dm}{d{x}_1}=\frac{x_1}{3(\sqrt{x_1^2-1})}\text{if}{x}_1>1$