/*! 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 23 Identify the conic and sketch it... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 23

Identify the conic and sketch its graph. \(r=\frac{3}{2+4 \sin \theta}\)

Short Answer

Expert verified
The conic section given by the equation \(r=\frac{3}{2+4 \sin \theta}\) is a hyperbola. After converting to cartesian coordinates, the graph can be sketched accordingly. The graph passes through x = 0 and x = 2, and has two asymptotes at x = 2 and y = 0.

Step by step solution

01

Identify the Conic

By looking at the given equation, it's not immediately clear which conic section this equation represents. Let's convert this polar coordinate equation to cartesian coordinate form first. From polar to cartesian, we know that \(x= r \cos \theta\) and \(y = r \sin \theta\), where \(r\) is the distance from origin, and \(\theta\) is the angle with respect to the positive X-axis. Replace \(r\) in both of the equations to get cartesian form.
02

Convert to Cartesian Coordinates

First, find \(x\): \(x = \frac{3\cos \theta}{2+4\sin \theta}\). Similarly, find \(y\): \(y = \frac{3 \sin \theta}{2+4\sin \theta}\). Now divide \(y/x\) to eliminate \(\theta\): \(y/x = \tan \theta = \sin \theta/\cos \theta\). This gives us \(y = \frac{x}{2-x}\).
03

Identify and Sketch Conic

The equation \(y = \frac{x}{2-x}\), is the equation of a downward opening hyperbola. The conic can be sketched based on this equation. The graph will pass through the points x = 0 and x = 2, with two asymptotes, one at x = 2 and one at y = 0. As x approaches to 2 from the left, y approaches to negative infinity.

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

The equations of a parabola and a tangent line to the parabola are given. Use a graphing utility to graph both equations in the same viewing window. Determine the coordinates of the point of tangency. \(x^{2}+12 y=0 \quad x+y-3=0\)

Determine whether the statement is true or false. Justify your answer. It is possible for a parabola to intersect its directrix.

Find the standard form of the equation of the ellipse with the given characteristics and center at the origin. Foci: (卤5,0)\(;\) major axis of length 14

The concept of ________ is used to measure the ovalness of an ellipse.

Find the standard form of the equation of the ellipse with the given characteristics. Center: (2,-1)\(;\) vertex: \(\left(2, \frac{1}{2}\right) ;\) minor axis of length 2
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Geometry

Read Explanation

Logic and Functions

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.