/*! 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 17 In Exercises 15-28, identify the... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 17

In Exercises 15-28, identify the conic and sketch its graph. \(r=\dfrac{5}{1+\sin\ \theta}\)

Short Answer

Expert verified
The equation \(r = \dfrac{5}{1+\sin\ \theta}\) represents a parabola with focus at the origin. The Cartesian representation of the polar equation is \(x = 5\) and \(y = 5 - 2\sqrt{5(x - 5)}\). In the graph, the parabola will be opening downward and has the vertex at (5,5).

Step by step solution

01

Identify the conic section

First, we need to identify the conic section in this problem. Using the general polar equation of conic section, we could note that it has the form \(r = \dfrac{p}{1+e\sin\ \theta}\), with \(p = 5\), and \(e = 1\). Therefore, since \(e = 1\), it is a parabola.
02

Translate polar equation into Cartesian coordinates

A parabola with a horizontal axis in a Cartesian system is represented as \((x-h)^2 = 4p(y-k)\). Since we know \(r = \sqrt{x^2 + y^2}\) and \(y = r\sin\ \theta\), we can substitute \(r = \dfrac{5}{1+\sin\ \theta}\) into these equations to find x and y. After substitution, we find that our parabola in Cartesian coordinates is \(x = 5\) and \(y = 5 - 2\sqrt{5(x - 5)}\).
03

Sketch the Graph

Sketch the parabola using the equation from Step 2. The parabola is opening downward with vertex at (5,5).

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.

Conic Sections
Conic sections are the curves obtained by intersecting a right circular cone with a plane. They include circles, ellipses, parabolas, and hyperbolas, each defined by the intersection's angle relative to the cone's axis. For instance, when a plane cuts the cone parallel to its side, it results in a parabola, which is our focus in the discussed exercise. These shapes have important properties and applications, from the orbits of planets (ellipses) to the focus of satellite dishes (parabolas). Understanding the equation of a conic in polar form, like \( r=\frac{5}{1+\sin \theta} \), is essential in identifying and graphing these unique curves.

Polar Equations
Polar equations express the relationship between the radius \( r \) and the angle \( \theta \) in the polar coordinate system. These coordinates are based on angles and distances from a fixed point, unlike Cartesian coordinates, which use a grid. In the given exercise, \( r=\frac{5}{1+\sin \theta} \) is a polar equation representing a conic section. Each type of conic has a distinct equation form in polar coordinates, which relies on the directrix, eccentricity, and focus to define its shape and orientation.

Graph Sketching
Graph sketching involves plotting points or drawing curves on a coordinate plane to visualize mathematical equations. The process requires understanding the form of the equation and how its parameters change the graph's shape and position. For the parabola's equation derived in our exercise, \( x=5 \) and \( y=5-2\sqrt{5(x-5)} \) graph sketching can be carried out step by step鈥攊dentify key features such as the vertex, axis, and direction of opening before plotting. Effective graph sketching aids in comprehending a curve's behavior and is a fundamental skill in Algebra and Calculus.
Parabola
A parabola is a symmetrical open plane curve formed by the intersection of a cone with a plane parallel to its side. The defining feature of a parabola is that each point is equidistant from a fixed point, called the focus, and a fixed straight line, the directrix. The general polar equation of a parabola where the conic's focus is at the pole is \( r=\frac{p}{1+e\sin \theta} \), where \( e \) is the eccentricity, and \( p \) is the focal parameter. With \( e=1 \) in our example, the equation represents a parabola. Its properties make parabolas useful in optics, engineering, and physics.

Cartesian Coordinates
Cartesian coordinates are used to uniquely identify points on a plane using two numbers \( (x, y) \) corresponding to the horizontal and vertical distances from a fixed point called the origin. Converting polar equations like \( r=\frac{5}{1+\sin \theta} \) into Cartesian coordinates involves algebraic manipulation, which allows us to analyze the conic's behavior using a different mathematical perspective. For example, in the Cartesian conversion of our exercise, we observed how the parabola opens and where its vertex lies. This transition between coordinate systems is an integral part of understanding and utilizing mathematical concepts across various fields of study.

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 23-48, sketch the graph of the polar equation using symmetry, zeros, maximum \(r\)-values, and any other additional points. \(r=1 - 2\ \sin\ \theta)\)

In Exercises 23-48, sketch the graph of the polar equation using symmetry, zeros, maximum \(r\)-values, and any other additional points. \(r= 2\ \sec\ \theta\)

In Exercises 39-54, find a polar equation of the conic with its focus at the pole. \(\textit{Conic}\) Parabola \(\textit{Vertex or Vertices}\) \((5, \pi)\)

In Exercises 15-28, identify the conic and sketch its graph. \(r=\dfrac{6}{1+\cos\ \theta}\)

In Exercises 15-28, identify the conic and sketch its graph. \(r=\dfrac{2}{2+3\sin\ \theta}\)
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

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.