SATHEE: Hyperbola 2 Question 11

Hyperbola 2 Question 11

11.

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 $△ P M N$ , then the correct expression(s) is/are

(2015 Adv.)

(a) $\frac{d l}{d x_{1}} = 1 - \frac{1}{3 x_{1}^{2}}$ for $x_{1} > 1$

(b) $\frac{d m}{d x_{1}} = \frac{x_{1}}{3 (\sqrt{x_{1}^{2} - 1})}$ for $x_{1} > 1$

(d) $\frac{d m}{d y_{1}} = \frac{1}{3}$ for $y_{1} > 0$

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Answer:

Correct Answer: 11. (a, b, d)

Solution:

Equation of family of circles touching hyperbola at $(x_{1}, y_{1})$ is ${(x - x_{1})}^{2} + {(y - y_{1})}^{2} + λ (x x_{1} - y y_{1} - 1) = 0$

Now, its centre is $(x_{2}, 0)$ .

$\begin{array}{ll} ∴ & \frac{- (λ x_{1} - 2 x_{1})}{2}, \frac{- (- 2 y_{1} - λ y_{1})}{2} = (x_{2}, 0) \\ \Rightarrow & 2 y_{1} + λ y_{1} = 0 \Rightarrow λ = - 2 \\ and & 2 x_{1} - λ x_{1} = 2 x_{2} \Rightarrow x_{2} = 2 x_{1} \end{array}$

$∴ P (x_{1}, \sqrt{x_{1}^{2} - 1})$ and $N (x_{2}, 0) = (2 x_{1}, 0)$

As tangent intersect $X$ -axis at $M \frac{1}{x_{1}}, 0$ .

Centroid of $△ P M N = (l, m)$

$\Rightarrow \frac{3 x_{1} + \frac{1}{x_{1}}}{3}, \frac{y_{1} + 0 + 0}{3} = (l, m)$

$\Rightarrow l = \frac{3 x_{1} + \frac{1}{x_{1}}}{3}$

On differentiating w.r.t. $x_{1}$ , we get $\frac{d l}{d x_{1}} = \frac{3 - \frac{1}{x_{1}^{2}}}{3}$

$\Rightarrow \frac{d l}{d x_{1}} = 1 - \frac{1}{3 x_{1}^{2}}$ , for $x_{1} > 1$ and $m = \frac{\sqrt{x_{1}^{2} - 1}}{3}$

On differentiating w.r.t. $x_{1}$ , we get

Also,

$\frac{d m}{d x_{1}} = \frac{2 x_{1}}{2 \times 3 \sqrt{x_{1}^{2} - 1}} = \frac{x_{1}}{3 \sqrt{x_{1}^{2} - 1}} for x_{1} > 1$

On differentiating w.r.t. $y_{1}$ , we get $\frac{d m}{d y_{1}} = \frac{1}{3}$ , for $y_{1} > 0$

Hyperbola 2 Question 11

11.

Answer:

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Hyperbola 2 Question 11

11.

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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