/*! 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 62 Convert the polar equation to re... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 62

Convert the polar equation to rectangular form. $$r=\frac{1}{1-\cos \theta}$$

Short Answer

Expert verified
So, the rectangular form of the polar equation \( r=\frac{1}{1-\cos \theta} \) is \( y^2 = 2x +1 \)

Step by step solution

01

Rewrite the equation using the substitution

Let's start by correlating the polar coordinates to rectangular coordinates. We'll use the conversion formulas \( x = r \cos \theta \) and \( y = r \sin \theta \). So we remake our given equation in terms of x as: \( r = \frac{1}{1 - \cos \theta} = \frac{1}{1 - \frac{x}{r}} \)
02

Simplify the equation by clearing the fraction

By cross multiplying(r* (1-\( \frac{x}{r} \)) = 1), the equation becomes \( r - x = 1 \) and after rearranging we get \( r = x + 1 \)
03

Substitution of r

We know that \( r^2 = x^2 + y^2 \) in polar coordinates is equivalent to \( r = \sqrt{x^2 + y^2} \) in rectangular coordinates, where r is the distance from the origin to the point (x,y). Therefore substituting \( x + 1 \) for \( r \) gives us the rectangular form of the equation as \(x^2 + y^2 = (x+1)^2 \)
04

Simplification

Simplify further and solve the equation to the simple form by expanding the square: \( x^2 + y^2 = x^2 + 2x +1 \). On simplifying, the equation becomes \( y^2 = 2x +1 \)

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

In Exercises 33-48, find a polar equation of the conic with its focus at the pole. $$ \begin{array}{ll} {\text { Conic }} & \text { Vertex or Vertices } \\ \text { Parabola } & (6,0) \\ \end{array} $$

Foci: \((3,2),(3,-4)\); major axis of length 8

\(r=2 \cos \left(\frac{3 \theta}{2}\right)\)

Use a graphing utility to graph and identify \(r=2+k \sin \theta\) for \(k=0,1,2\), and 3 .

In Exercises 7-12, test for symmetry with respect to \(\theta=\pi / 2\), the polar axis, and the pole. \(r=5+4 \cos \theta\)
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Statistics

Read Explanation

Logic and Functions

Read Explanation

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Geometry

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.