Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 13: Problem 1105

The order of the differential equation of family of circle touching a fixed straight line passing through origin is. (A) 2 (B) 3 (C) 4 (D) none of these

Short Answer

Expert verified
The order of the differential equation of the family of circles touching a fixed straight line passing through the origin is \(2\). Hence, the answer is (A) 2.

Step by step solution

01

Write the equation of a circle

A circle with center (h, k) and radius r is given by \((x-h)^2 + (y-k)^2 = r^2\). Now, let's derive the equation of the straight line passing through the origin.
02

Determine the equation of the straight line passing through the origin

Let the fixed straight line passing through the origin have a slope of m. The equation is given by \(y = mx\). Next, we will find the condition that the circle touches the straight line. This means that at some point on the circle, the line is tangent to the circle.
03

Find the condition for the line to touch the circle

The line touches the circle when the distance from the center of the circle to the line is equal to the radius, so we have \(r = \frac{|mk - h|}{\sqrt{1 + m^2}}\). Now, we will eliminate the arbitrary constants, (h, k) and r, from the circle equation.
04

Eliminate the arbitrary constants

Replace r in the circle equation with the expression in Step 3: \[(x - h)^2 + (y - k)^2 = \left(\frac{|mk - h|}{\sqrt{1 + m^2}}\right)^2\] We can use the equation of the line to replace k with k = y/mx in the above equation. After eliminating the constants and simplifying the equation, we will differentiate the resulting equation to obtain the differential equation.
05

Differentiate and find the order

Differentiate the equation obtained in Step 4 with respect to x once: \(2(x-h) + 2(y-k)\frac{dy}{dx} = 0\) Now differentiate again the latter equation once: \(\frac{d^2y}{dx^2} = \frac{1}{(k - y)^2(1 + m^2)}\) Since the equation now only contains second-order derivatives, the order of the differential equation is 2. Hence, the answer is (A) 2.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The solution of \((\mathrm{dy} / \mathrm{dx})=4 \mathrm{x}+\mathrm{y}+1 \mathrm{is}:\) (A) \(4 \mathrm{x}+\mathrm{y}+1=\mathrm{c} \cdot \mathrm{e}^{\mathrm{x}}\) (B) \(4 \mathrm{x}+\mathrm{y}+5=\mathrm{e}^{\mathrm{x}}+\mathrm{c}\) (C) \(4 \mathrm{x}+\mathrm{y}+5=\mathrm{c} \cdot \mathrm{e}^{\mathrm{x}}\) (D) none of these

Solution of \((\mathrm{dy} / \mathrm{dx})=1+\mathrm{x}+\mathrm{y}^{2}+\mathrm{xy}^{2}, \mathrm{y}(0)=0\) is: (A) \(y=\tan \left(c+x+x^{2}\right)\) (B) \(\mathrm{y}=\tan \left[\mathrm{x}+\left(\mathrm{x}^{2} / 2\right)\right]\) (C) \(\mathrm{y}^{2}=\exp \left[\mathrm{x}+\left(\mathrm{x}^{2} / 2\right)\right]\) (D) \(\mathrm{y}^{2}=1+\mathrm{c} \cdot \exp \left[\mathrm{x}+\left(\mathrm{x}^{2} / 2\right)\right]\)

Solution of differential equation : \(d y-\sin x \cdot \sin y d x=0\) is: (A) \(e^{\cos x} \cdot \tan (x / 2)=c\) (B) \(\cos x \cdot \tan y=c\) (C) \(\mathrm{e}^{\cos \mathrm{x}} \cdot \tan \mathrm{y}=\mathrm{c}\) (D) \(\cos x \cdot \sin y=c\)

The differential equation of family of circles of radius 'a' is: (A) \(a^{2} y_{2}=\left[1-y_{1}^{3}\right]^{2}\) (B) \(a^{2} y_{2}=\left[1-y_{1}^{2}\right]^{3}\) (C) \(a^{2}\left(y_{2}\right)^{2}=\left[1+y_{1}^{3}\right]^{2}\) (D) \(a^{2}\left(y_{2}\right)^{2}=\left[1+y_{1}^{2}\right]^{3}\)

The solution of \(\mathrm{xdy}-\mathrm{ydx}=0\) represents: (A) parabola having vertex at \((0,0)\) (B) circle having centre at \((0,0)\) (C) a straight line passing through \((0,0)\) (D) a rectangular hyperbola
See all solutions

Recommended explanations on English Textbooks

Semiotics

Read Explanation

Essay Plan

Read Explanation

Research and Composition

Read Explanation

Morphology

Read Explanation

Lexis and Semantics Summary

Read Explanation

English Grammar Summary

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free