Question

Consider the Hyperbola $H:x^2-y^2=1$ and 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 $△PMN$, then the correct expression(s) is (are):

• A. $\frac{dl}{d{x}_1}=1-\frac{1}{3x^2}$ for $x_1>1$
• B. $\frac{dm}{d{x}_1}=\frac{x_1}{3\sqrt{x_1^2-1}}$ for $x_1>1$
• C. $\frac{dl}{d{x}_1}=1+\frac{1}{3x^2}$ for $x_1>1$
• D. $\frac{dm}{d{y}_1}=\frac{1}{3}$ for $y_1>0$

Answer 1

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

$\frac{dl}{d{x}_1}=1-\frac{1}{3x^2}$ for $x_1>1$
$\frac{dm}{d{x}_1}=\frac{x_1}{3\sqrt{x_1^2-1}}$ for $x_1>1$
$\frac{dm}{d{y}_1}=\frac{1}{3}$ for $y_1>0$

Tangent at $(x_1,y_1)$ to the hyperbola $x^2-y^2=1$ is $x{x}_1-y{y}_1=1$ -----(1)
Tangent at $(x_1,y_1)$ to the circle $(x-x_2{)}^2+y^2=(x_1-x_2{)}^2+y_1^2$ is
$(x_1-x_2)x+y_{1}y=x_1^2-x_1{x}_2+y_1^2$ -----(2)
comparing (2) with (1) gives
$\frac{x_1-x_2}{x_1}=\frac{y_1}{-y_1}=\frac{x_1^2-x_1{x}_2+y_1^2}{1}$
$\Rightarrow x_1-x_2=-x_1$ and $x_1^2-x_1{x}_2+y_1^2=-1$
$\Rightarrow 2x_1=x_2$
Let the tangent (1) cut the x-axis at $M\left(\frac{1}{x_1},0\right)$
centroid of $\Delta PMN=\left(\frac{x_1+x_2+x_3}{3},\frac{y_1}{3}\right)=(l,m)$
$\Rightarrow x_1+x_2+x_3=3l$ and $y_1=3m$
$\Rightarrow x_1+2x_1+\frac{1}{x_1}=3l$ and $y_1=3m$
$\Rightarrow l=x_1+\frac{1}{3x_1}$ and $m=\frac{y_1}{3}$
$\therefore \frac{dl}{d{x}_1}=1-\frac{1}{3x_1^2}$
$\frac{dm}{d{x}_1}=\frac{1}{3}\frac{d{y}_1}{d{x}_1}=\frac{1}{3}\frac{d\sqrt{x_1^2-1}}{d{x}_1}$
$\therefore \frac{dm}{d{x}_1}=\frac{x_1}{3\sqrt{x_1^2-1}}$
$\therefore \frac{dm}{d{y}_1}=\frac{1}{3}$
Hence, options A,B and D.