Question

Consider th 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 $x_1>and{y}_1>0$ the common tangent to $H$ and $S$ at $P$ intersects the X-axis at point M. If $(l,m)$ is the centroid of $\Delta PMN$; then the correct expression (s) is

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

Answer 1

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

$\frac{dm}{d{y}_1}=\frac{1}{3}for{y}_1>0$


Equation of family of circles touching hyperbola $(x_1,y_1)is(x-x_1{)}^2+(y-y_1{)}^2+\lambda (x{x}_1-y{y}_1-1)=0$
Now its centre is $(x_2,0).$
$\therefore [\frac{-(\lambda x_1-2x_1)}{2},\frac{-(-2y_1-\lambda y_1)}{2}]=(x_2,0)\Rightarrow 2y_1+\lambda y_1=0\Rightarrow \lambda =-2and2x_1-\lambda x_1=2x_2\Rightarrow x_2=2x_1\therefore P(x_1,\sqrt{x_1^2-1})andN(x_2,0)=(2x_1,0)$
As tangent intersect X-axis at$M(\frac{1}{x_1},0).$
Centroid of $\Delta PMN=(l,m)$
$(\frac{3x_1+\frac{1}{x_1},}{3},\frac{y_1+0+0}{3})=(l,m)\Rightarrow l=\frac{3x_1+\frac{1}{x_1}}{3}$
On differentiating w.r.t. $x_1$ we get $\frac{dl}{d{x}_1}=\frac{3-\frac{1}{x_1^2}}{3}\Rightarrow \frac{dl}{d{x}_1}=1-\frac{1}{3x_1^2},for{x}_1>1andm=\frac{\sqrt{x_1^2-1}}{3}$
On differentiating w.r.t. $x_1,$ get
$\frac{dm}{d{x}_1}=\frac{2x}{2\times 3\sqrt{x_1^2-1}}=\frac{x_1}{3\sqrt{x_1^2-1}}for{x}_1>1$
Also , $m=\frac{y_1}{3}$
on differentiating w.r.t. $y_1$. we get $\frac{dm}{d{y}_1}=\frac{1}{3},$ for $y_1>0$