/*! 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 31 Eliminate the parameter and obta... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 31

Eliminate the parameter and obtain the standard form of the rectangular equation. Ellipse: \(x=h+a \cos \theta, y=k+b \sin \theta\)

Short Answer

Expert verified
The standard form of the rectangle equation is \( \left( \frac{x-h}{a} \right)^2 + \left( \frac{y-k}{b} \right)^2 = 1 \).

Step by step solution

01

Express the Parametric Equations by Isolating Cosine and Sine

Write the given equations in terms of \( \cos \theta \) and \( \sin \theta \). With \( x=h+a \cos \theta \) => \( \frac{x-h}{a} = \cos \theta \)And with \( y=k+b \sin \theta \) => \( \frac{y-k}{b} = \sin \theta \)
02

Apply Pythagorean Identity

Apply the Pythagorean Identity to get the rectangular form of the equations.According to the Pythagorean Identity, \( \cos^2 \theta + \sin^2 \theta = 1\).So replace \( \cos \theta \) with \(\frac{x-h}{a}\) and \( \sin \theta \) with \(\frac{y-k}{b}\).So, \( \left( \frac{x-h}{a} \right)^2 + \left( \frac{y-k}{b} \right)^2 = 1 \)
03

Obtain the final form of the equation

Now, expand the equation to get the final rectangular form of the equation \( \left( \frac{x-h}{a} \right)^2 + \left( \frac{y-k}{b} \right)^2 = 1 \). This is the standard form of the equation for an ellipse.

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

Find the standard form of the equation of the ellipse with the given characteristics and center at the origin. Foci: (卤2,0)\(;\) major axis of length 10

The relation between the velocity \(y\) (in radians per second) of a pendulum and its angular displacement \(\theta\) from the vertical can be modeled by a semiellipse. A 12-centimeter pendulum crests \((y=0)\) when the angular displacement is -0.2 radian and 0.2 radian. When the pendulum is at equilibrium \((\theta=0),\) the velocity is -1.6 radians per second. (a) Find an equation that models the motion of the pendulum. Place the center at the origin. (b) Graph the equation from part (a). (c) Which half of the ellipse models the motion of the pendulum?

Use a graphing utility to graph the ellipse. Find the center, foci, and vertices. (Recall that it may be necessary to solve the equation for \(y\) and obtain two equations.) \(36 x^{2}+9 y^{2}+48 x-36 y-72=0\)

The chord perpendicular to the major axis at the center of the ellipse is called the _________ ____________ of the ellipse.

Find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin. Vertical axis and passes through the point (-3,-3)
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation

Calculus

Read Explanation

Decision Maths

Read Explanation

Applied Mathematics

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.