SATHEE: Differential Equations 1 Question 18

Differential Equations 1 Question 18

18. Let $y = f (x)$ be a curve passing through $(1, 1)$ such that the triangle formed by the coordinate axes and the tangent at any point of the curve lies in the first quadrant and has area 2 unit. Form the differential equation and determine all such possible curves.

$(1995, 5$ M)

Integer Answer Type Question

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Answer:

Correct Answer: 18. (b)

Solution:

Equation of tangent to the curve $y = f (x)$ at point

whose, $x$ -intercept $x - y \cdot \frac{d x}{d y}, 0$

$y -intercept 0, y - x \frac{d y}{d x}$

Given, $△ O P Q = 2$

$\Rightarrow \frac{1}{2} \cdot x - y \frac{d x}{d y} y - x \frac{d y}{d x} = 2$

$\Rightarrow x - y \frac{1}{p} (y - x p) = 4, where p = \frac{d y}{d x}$

$\Rightarrow p^{2} x^{2} - 2 p x y + 4 p + y^{2} = 0$

$\Rightarrow (y - p x)^{2} + 4 p = 0$

$∴ y - p x = 2 \sqrt{- p}$

$\Rightarrow y = p x + 2 \sqrt{- p}$

On differentiating w.r.t. $x$ , we get

$\begin{aligned} p = p + \frac{d p}{d x} \cdot x + 2 \cdot \frac{1}{2} (- p)^{- 1 / 2} \cdot (- 1) \frac{d p}{d x} \\ \Rightarrow & \frac{d p}{d x} x - (- p)^{- 1 / 2} = 0 \\ \Rightarrow & \frac{d p}{d x} & = 0 or x = (- p)^{- 1 / 2} \\ If & \frac{d p}{d x} & = 0 \Rightarrow p = c \end{aligned}$

On putting this value in Eq. (i), we get $y = c x + 2 \sqrt{- c}$

This curve passes through $(1, 1)$ .

$\begin{aligned} \Rightarrow & 1 & = c + 2 \sqrt{- c} \\ \Rightarrow & c & = - 1 \\ ∴ & y & = - x + 2 \\ \Rightarrow & x + y & = 2 \end{aligned}$

Again, if $x = (- p)^{- 1 / 2}$

$\Rightarrow - p = \frac{1}{x^{2}}$ putting in Eq. (i)

$y = \frac{- x}{x^{2}} + 2 \cdot \frac{1}{x} \Rightarrow x y = 1$

Thus, the two curves are $x y = 1$ and $x + y = 2$ .

Differential Equations 1 Question 18

18. Let $y = f (x)$ be a curve passing through $(1, 1)$ such that the triangle formed by the coordinate axes and the tangent at any point of the curve lies in the first quadrant and has area 2 unit. Form the differential equation and determine all such possible curves.

Integer Answer Type Question

Answer:

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Differential Equations 1 Question 18

18. Let y=f(x) be a curve passing through (1,1) such that the triangle formed by the coordinate axes and the tangent at any point of the curve lies in the first quadrant and has area 2 unit. Form the differential equation and determine all such possible curves.

Integer Answer Type Question

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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18. Let $y = f (x)$ be a curve passing through $(1, 1)$ such that the triangle formed by the coordinate axes and the tangent at any point of the curve lies in the first quadrant and has area 2 unit. Form the differential equation and determine all such possible curves.