/*! 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 282 For the following exercises, con... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 282

For the following exercises, consider the following polar equations of conics. Determine the eccentricity and identify the conic. \(r=\frac{5}{-1+2 \sin \theta}\)

Short Answer

Expert verified
Eccentricity is 2; conic is a hyperbola.

Step by step solution

01

Understanding the Polar Equation

The given polar equation is \( r = \frac{5}{-1 + 2 \sin \theta} \). This equation is in the form \( r = \frac{ed}{1 - e \sin \theta} \) for conics where the directrix is vertical with \( \theta = 90^\circ \). Our job is to identify the eccentricity \( e \) and determine the type of conic it represents.
02

Comparing with the Standard Form

To find the eccentricity, compare \( r = \frac{5}{-1 + 2 \sin \theta} \) with \( r = \frac{ed}{1 - e \sin \theta} \). Here, \( e = 2 \) and \( ed = 5 \), so \( d \) can be calculated as \( d = \frac{5}{2} \).
03

Determine the Eccentricity

From the comparison, we find that the eccentricity \( e \) is 2. The eccentricity is a key feature in determining the type of conic.
04

Identifying the Conic Type

Based on conic properties: - If \( e = 1 \), it's a parabola.- If \( e < 1 \), it's an ellipse.- If \( e > 1 \), it's a hyperbola.Since \( e = 2 > 1 \), the conic is identified as a hyperbola.

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.

Eccentricity
Eccentricity is a crucial concept when studying conic sections. It is a measure that describes how much a conic section deviates from being a perfect circle. For any conic section represented by a polar equation, the eccentricity is denoted as \(e\). The value of \(e\) can determine the shape and type of conic section:
  • If \(e < 1\), the conic is an ellipse.
  • If \(e = 1\), the conic is a parabola.
  • If \(e > 1\), the conic is a hyperbola.
Let's use these relationships to determine conic shapes. Eccentricity is like a fingerprint for conics, making it easy to identify their specific type. Familiarizing yourself with how to calculate \(e\) directly from the conic's equation can greatly aid in quickly recognizing the conic's nature and characteristics.
Conic Sections
Conic sections arise from slicing a three-dimensional cone with a plane, and each unique slice forms a different conic:
  • An ellipse is formed when the plane intersects the cone at an angle, creating an oval shape.
  • A parabola occurs when the plane is parallel to the slope of the cone.
  • A hyperbola results when the plane slices through both nappes (the two separate pieces) of the cone.
Each of these shapes can be represented by specific equations in various coordinate systems, including polar coordinates. Understanding how changes in the angle and position of the cutting plane affect the resulting conic can enrich your understanding of their properties and applications in fields such as physics and engineering.
Hyperbola
A hyperbola is one type of conic section characterized by its two divergent branches, giving it a distinct open curve appearance. When the eccentricity \(e\) of a conic section satisfies \(e > 1\), the conic is a hyperbola. This is evident in the exercise, where \(e = 2\). Hyperbolas have intriguing properties:
  • They have two branches mirrored across their center.
  • Each branch approaches a pair of asymptotes but never touches them.
  • The foci of a hyperbola, like in ellipses, are points from which the difference in distances to any point on the hyperbola is constant.
These characteristics make hyperbolas important in various real-world applications, such as in the paths of certain celestial bodies and in the design of certain optical devices.
Polar Coordinates
Polar coordinates offer a different perspective from Cartesian coordinates, especially useful when dealing with curves like spirals and conics. In this system, each point is determined by a distance \(r\) from a fixed point, and an angle \(\theta\) from a fixed direction. The polar equation for conics typically takes a form such as \(r = \frac{ed}{1 - e \cos \theta}\) or \(r = \frac{ed}{1 - e \sin \theta}\), depending on the position of the directrix.
  • This coordinate system is especially handy for problems involving rotational symmetry.
  • It allows for easier integration of angular relationships in equations.
  • Transitioning between Cartesian and polar coordinates is often crucial for calculus problems.
Mastering polar coordinates can simplify many problems and provides a richer understanding of geometric shapes and their properties.

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

For the following exercises, find the slope of the tangent line to the given polar curve at the point given by the value of \(\theta\) [T] Use technology: \(r=2+4 \cos \theta\) at \(\theta=\frac{\pi}{6}\)

For the following exercises, find the area of the described region. Enclosed by \(r=1+\sin \theta\)

For the following exercises, find the slope of the tangent line to the given polar curve at the point given by the value of \(\theta\) \(r=3 \cos \theta, \theta=\frac{\pi}{3}\)

For the following exercises, determine the equation of th parabola using the information given. Focus \((4,0)\) and directrix \(x=-4\)

For the following exercises, determine the polar equation form of the orbit given the length of the major axis and eccentricity for the orbits of the comets or planets. Distance is given in astronomical units (AU). Halley's Comet: length of major axis \(=35.88\), eccentricity \(=0.967\)
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

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.