/*! 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 65 Convert the rectangular equation... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 65

Convert the rectangular equation to polar form. Assume \(a>0\). \(x^{2}+y^{2}=9\)

Short Answer

Expert verified
The polar form of the equation \(x^{2}+y^{2}=9\) is \(r^{2}=9\).

Step by step solution

01

Transform Rectangular to Polar Equations

We start with the standard rectangular equation \(x^{2}+y^{2}=9\). In polar form, we replace \(x^{2} + y^{2}\) with \(r^{2}\) according to the relationship \(x^{2} + y^{2} = r^{2}\). Here, \(r^{2}\) represents the square of the distance from the origin and theta is the angle formed with the positive x-axis. Replace this in our equation we get, \(r^{2}=9\).

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!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rectangular to Polar Conversion
When working with equations, it's essential to understand how to translate coordinates from one system to another. Rectangular coordinates
  • Use the orthogonal axes, x and y, making them intuitive for everyday horizontal and vertical measurements.
  • Consist of equations often characterized by variables like x and y in formulas such as lines, circles, or parabolas.
But sometimes, converting these to polar coordinates is beneficial. Polar coordinates focus on distances from a point and angles, which can simplify the math for equations dealing with rotations or circular patterns.
  • To convert, start from the relationship between the two systems:
    • \( x = r \cos(\theta) \)
    • \( y = r \sin(\theta) \)
  • When an equation like \( x^2 + y^2 = 9 \) appears in rectangular form, recognize it equates to \( r^2 = 9 \) in polar form.
This transformation simplifies several geometric interpretations, allowing a different perspective on the equation.
Rectangular Equations
Rectangular equations describe relationships using the familiar x and y axes. These frameworks are prevalent in algebra and geometry due to their straightforward nature. Equations like \( x^2 + y^2 = 9 \) are often circles in this system, where
  • The equation itself defines a circle with a radius, calculated from the equation in place.
  • The points on the circle satisfy conditions derived from squares of distances along the x and y axes.
This form makes evaluating intersections, slopes, and other algebraic properties more manageable.
However, converting to polar facilitates handling equations involving angles or curves, offering a simplified view for some complex relationships.
Polar Form
In polar form, the equation focuses on relationships built around angles and radii. This is particularly useful when dealing with rotations, oscillations, and angle-based calculations.
  • A circle such as depicted by \( r^2 = 9 \) clarifies instantly as a circle centered at the origin with radius 3.
  • It uses fundamental polar formulas:
    • \( r = \sqrt{x^2 + y^2} \)
    • \( \theta = \tan^{-1}(y/x) \)
This view not only shows that you don't need x and y values explicitly but visualizes symmetry and radial properties natively.
The polar form can change the question of position into a question of radius and angle, often streamlining complex problems by simplifying the equation into a more intuitive radial format.

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

Explain what each of the following equations represents, and how equations (a) and (b) are equivalent. (a) \(y=a(x-h)^{2}+k, \quad a \neq 0\) (b) \((x-h)^{2}=4 p(y-k), \quad p \neq 0\) (c) \((y-k)^{2}=4 p(x-h), \quad p \neq 0\)

Find the standard form of the equation of the parabola with the given characteristics. Vertex: (0,2)\(;\) directrix: \(y=4\)

Find the standard form of the equation of the ellipse with the given characteristics. Foci: (0,0),(0,8)\(;\) major axis of length 16

Find the vertex, focus, and directrix of the parabola, and sketch its graph. \(y=\frac{1}{4}\left(x^{2}-2 x+5\right)\)

Find the standard form of the equation of the ellipse with the given characteristics. Center: (0,4)\(; a=2 c ;\) vertices: (-4,4),(4,4)
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

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.