/*! 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 28 Convert the polar equation of a ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 28

Convert the polar equation of a conic section to a rectangular equation. $$ r(2.5-2.5 \sin \theta)=5 $$

Short Answer

Expert verified
The rectangular equation is \( x^2 + y^2 = 1 \).

Step by step solution

01

Expand and Rearrange the Polar Equation

Start with the given polar equation: \[ r(2.5 - 2.5 \sin \theta) = 5 \]. Expand the left-hand side to get: \[ 2.5r - 2.5r \sin \theta = 5 \]. Isolate the term with \( r \sin \theta \) to obtain: \[ 2.5r \sin \theta = 2.5r - 5 \]. Divide through by 2.5 to simplify: \[ r \sin \theta = r - 2 \].
02

Replace Polar Coordinates with Rectangular Coordinates

Recall the conversion formulas: \( x = r \cos \theta \) and \( y = r \sin \theta \). Substitute \( y \) for \( r \sin \theta \) in the equation derived from Step 1: \[ y = r - 2 \]. Then, express \( r \) in terms of \( x \) and \( y \) using \( r = \sqrt{x^2 + y^2} \), yielding: \[ y = \sqrt{x^2 + y^2} - 2 \].
03

Square Both Sides

To eliminate the square root, square both sides of the equation from Step 2: \[ y^2 = (\sqrt{x^2 + y^2} - 2)^2 \]. This gives: \[ y^2 = x^2 + y^2 - 4\sqrt{x^2 + y^2} + 4 \].
04

Rearrange and Solve for a Rectangular Equation

Move \( x^2 + y^2 \) to the left side to combine like terms: \[ 0 = -4\sqrt{x^2 + y^2} + 4 \]. Solve for \( \sqrt{x^2 + y^2} \): \[ \sqrt{x^2 + y^2} = 1 \]. Square both sides to eliminate the square root: \[ x^2 + y^2 = 1 \].
05

Interpret the Result

The resulting equation \( x^2 + y^2 = 1 \) represents a circle in rectangular coordinates, with its center at the origin (0,0) and a radius of 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!

Key Concepts

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

Polar Coordinates
Polar coordinates offer a unique way to describe locations on a plane. Instead of using two perpendicular axes like rectangular coordinates, polar coordinates use a radius and an angle, which can sometimes simplify problems involving symmetry or circular motion. Here are the essentials to grasp:
  • The radius \( r \) is the distance from the origin to a point, while the angle \( \theta \) is measured from the positive x-axis.
  • A point in the plane is represented as \((r, \theta)\), which specifies its exact location.
  • Polar equations often describe curves that are naturally circular or spiral.
By understanding these basic elements, you gain a versatile mathematical tool useful in various applications, from navigation to engineering.
Conic Sections
Conic sections arise when a plane intersects a cone, giving us familiar shapes like circles, ellipses, parabolas, and hyperbolas. These are significant in both pure and applied mathematics.
  • A circle, the simplest form of a conic section, is seen here in the transition from polar coordinates to a rectangular form \( x^2 + y^2 = 1 \). This foregrounds a symmetrical relationship around a central point.
  • Ellipses and hyperbolas have interesting properties worth exploring but aren't directly applicable to this example.
  • Understanding the transformation of a conic section between coordinate systems aids in visualizing their structures and applications.
Conic sections expand our ability to model and solve real-world problems, from physics to graphics programming.
Rectangular Coordinates
Rectangular coordinates, also known as Cartesian coordinates, involve specifying a point's location in terms of two numbers \((x, y)\) along perpendicular axes. This system is intuitive and widely used for various reasons.
  • Equations like \( x^2 + y^2 = 1 \) reveal their geometrical nature, which is straightforward to interpret.
  • Transforming from polar to rectangular coordinates can simplify equations, making them easier to work with algebraically.
  • Such transformations emphasize the grid-like nature of the rectangular coordinate system, facilitating understanding of planar geometry.
Therefore, mastery of rectangular coordinates provides a strong foundation in mathematics that is essential for graphing functions and solving coordinate geometry problems.

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

Graph the given conic section. If it is a parabola, label the vertex, focus, and directrix. If it is an ellipse, label the vertices and foci. If it is a hyperbola, label the vertices and foci. $$ r(3-2 \sin \theta)=6 $$

For the following exercises, find the equation of the parabola given information about its graph. Vertex is (0,0)\(;\) directrix is \(y=4,\) focus is (0,-4) .

Express each equation in polar form with \(r\) as a function of \(\theta\). $$ 16 x^{2}+24 x y+9 y^{2}=4 $$

Express each equation in polar form with \(r\) as a function of \(\theta\). $$ x y=2 $$

For the following exercises, identify the conic with a focus at the origin, and then give the directrix and eccentricity. $$ r=\frac{3}{10+10 \cos \theta} $$
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Statistics

Read Explanation

Logic and Functions

Read Explanation

Pure Maths

Read Explanation

Calculus

Read Explanation

Theoretical and Mathematical Physics

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.