/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 11 Find the orthogonal trajectories... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 11

Find the orthogonal trajectories of the family of ellipses having center at the origin, a focus at the point \((c, 0)\), and semimajor axis of length \(2 c\).

Short Answer

Expert verified
The orthogonal trajectories of the family of ellipses with center at the origin, a focus at the point \((c, 0)\), and semimajor axis of length \(2c\) is given by the equation: \[y^4 = 8x^4\]

Step by step solution

01

Find the equation of the family of ellipses

To find the equation for the given family of ellipses, we will use the general equation of an ellipse centered at the origin: \[\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\] Since the semimajor axis has a length of \(2c\), we have \(a = 2c\). To find the length of semiminor axis \(b\), we will use the distance between the center of the ellipse and a focus, which is \(c\). The relationship between \(a\), \(b\), and \(c\) is given by the following equation: \[c^2 = a^2 - b^2\] Substitute \(a = 2c\) into the equation: \[c^2 = (2c)^2 - b^2\] Solve for \(b\): \[b^2 = (2c)^2 - c^2 = 3c^2\] Now, we have the equation of the family of ellipses: \[\frac{x^2}{(2c)^2}+\frac{y^2}{3c^2}=1\]
02

Differentiate the equation

Next, we will differentiate the equation with respect to \(x\) using implicit differentiation. The derivative is denoted as \(\frac{dy}{dx}\). Differentiate both sides while treating \(c\) as a constant: \[\frac{2x}{(2c)^2} \cdot \frac{dx}{dx} + \frac{2y}{3c^2} \cdot \frac{dy}{dx} = 0\] Simplify the equation and solve for \(\frac{dy}{dx}\): \[\frac{dy}{dx} = -\frac{2x}{y}\]
03

Replace the derivative with its negative reciprocal

To find the orthogonal trajectories, we will replace \(\frac{dy}{dx}\) with its negative reciprocal, \(-\frac{y}{2x}\): \[-\frac{y}{2x} = -\frac{2x}{y}\]
04

Solve the new equation for \(y\)

Now, we will solve the new equation for \(y\): \[\frac{y^2}{2x} = \frac{4x^3}{y^2}\] Multiply both sides by \(y^2\): \[y^4 = 8x^4\]
05

Write the final equation

The final equation for the orthogonal trajectories is: \[y^4 = 8x^4\] This equation represents the orthogonal trajectories of the family of ellipses with center at the origin, a focus at the point \((c, 0)\), and semimajor axis of length \(2c\).

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.

Over 30 million students worldwide already upgrade their learning with 91影视!

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

A stone weighing \(4 \mathrm{lb}\) falls from rest toward the earth from a great height. As it falls it is acted upon by air resistance that is numerically equal to \(\frac{1}{2} v(\mathrm{in}\) pounds), where \(v\) is the velocity (in feet per second). (a) Find the velocity and distance fallen at time \(t\) sec. (b) Find the velocity and distance fallen at the end of 5 sec.

A ship which weighs 32,000 tons starts from rest under the force of a constant propeller thrust of \(100,000 \mathrm{lb}\). The resistance in pounds is numerically equal to \(8000 v\), where \(v\) is in feet per second. (a) Find the velocity of the ship as a function of the time. (b) Find the limiting velocity (that is, the limit of \(v\) as \(t \rightarrow+\infty\) ). (c) Find how long it takes the ship to attain a velocity of \(80 \%\) of the limiting velocity.

Assume that the rate of change of the human population of the earth is proportional to the number of people on earth at any time, and suppose that this population is increasing at the rate of \(2 \%\) per year. The 1979 World Almanac gives the 1978 world population estimate as 4219 million; assume this figure is in fact correct. (a) Using this data, express the human population of the earth as a function of time. (b) According to the formula of part (a), what was the population of the earth in \(1950 ?\) The 1979 World Almanac gives the 1950 world population estimate as 2510 million. Assuming this estimate is very nearly correct, comment on the accuracy of the formula of part (a) in checking such past populations. (c) According to the formula of part (a), what will be the population of the earth in 2000? Does this seem reasonable? (d) According to the formula of part (a), what was the population of the earth in \(1900 ?\) The 1970 World Almanac gives the 1900 world population estimate as 1600 million. Assuming this estimate is very nearly correct, comment on the accuracy of the formula of part (a) in checking such past populations. (e) According to the formula of part (a), what will be the population of the earth in 2100 ? Does this seem reasonable?

A man is pushing a loaded sled across a level field of ice at the constant speed of \(10 \mathrm{ft} / \mathrm{sec}\). When the man is halfway across the ice field, he stops pushing and lets the loaded sled continue on. The combined weight of the sled and its load is \(80 \mathrm{lb}\); the air resistance (in pounds) is numerically equal to \(\frac{3}{4} v\), where \(v\) is the velocity of the sled (in feet per second); and the coefficient of friction of the runners on the ice is \(0.04\). How far will the sled continue to move after the man stops pushing?

The human population of a certain island satisfies the logistic law \((3.58)\) with \(k=0.03, \lambda=3(10)^{-8}\), and time \(t\) measured in years. (a) If the population in 1980 is 200,000, find a formula for the population in future years. (b) According to the formula of part (a), what will be the population in the year \(2000 ?\) (c) What is the limiting value of the population at \(t \rightarrow \infty\) ?
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Probability and Statistics

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.