/*! 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 19 Graph and interpret the conic se... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 19

Graph and interpret the conic section. $$r=\frac{-6}{\sin (\theta-\pi / 4)-2}$$

Short Answer

Expert verified
Without having actual numerical coefficients, it's impossible to give a specific short answer for this question. However, the solution would involve converting the polar equation into rectangular form, identifying the type of conic, sketching the graph and providing interpretation based on the graphed conic section.

Step by step solution

01

Convert the Polar Equation into Rectangular Form

Start by expressing the polar coordinates \(r\) and \(\theta\) in terms of rectangular coordinates \(x\) and \(y\). The conversion relationships are \(x = r\cos\theta\) and \(y = r\sin\theta\). Plug these into the given polar equation, which will transform into a standard form of a conic section.
02

Identify the Conic Section

Inspect the equation obtained in Step 1. The coefficients and signs of the terms will reveal the type of conic section it represents - circle, ellipse, parabola, or hyperbola. As a tip, a circle's equation has square terms with equal coefficients, while for an ellipse they are unequal. Parabola has only one square term whereas a hyperbola has a subtraction between squares.
03

Graph the Conic Section

Sketch the graph of the identified conic section on a graph paper. Check if the graph fits all the conditions given in the converted equation, such as the center or the vertex, the axes, and the orientation.
04

Interpreting the Graph

Provide an interpretation of your graph. If it's a circle or an ellipse, discuss its radius or axes. If it's a parabola, describe its direction. If it's a hyperbola, note its vertices, foci, and asymptotes.

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 locate a point in the plane. Instead of using horizontal and vertical distances (as in rectangular coordinates), polar coordinates use a distance from a reference point and an angle from a reference direction.
  • The distance from the origin is denoted as \(r\).
  • The angle \(\theta\) is measured from the positive x-axis.
This approach is particularly useful for problems involving circular or spiral patterns, where specifying a radius and angle is more intuitive than using x and y coordinates. It's crucial to understand that in polar coordinates, any point can be described infinitely many ways. For instance, you can rotate through multiple full circles or choose a negative radius value, affecting the angle direction.
Rectangular Coordinates
Rectangular coordinates, also known as Cartesian coordinates, describe a point in terms of its horizontal and vertical distances from a fixed reference point, called the origin. The two components are denoted by \(x\) and \(y\).
  • \(x\) represents the horizontal distance along the x-axis.
  • \(y\) indicates the vertical distance along the y-axis.
These coordinates are commonly used because of their straightforward nature, making them ideal for most algebraic calculations and graphing. However, when dealing with circles or angles, they might not be as efficient as polar coordinates. Understanding how to switch between these coordinate systems enables more flexible problem-solving approaches.
Graphing Conics
Conic sections include circles, parabolas, ellipses, and hyperbolas. They arise from intersecting a plane with a double-napped cone. Each type of conic has distinct characteristics:
  • **Circle**: Equal coefficients in the squared terms and no linear term.
  • **Ellipse**: Unequal coefficients in the squared terms, both positive.
  • **Parabola**: Only one squared term present; can open in any direction.
  • **Hyperbola**: Squared terms with opposite signs.
When graphing conics, each has specific attributes that help in sketching: - The **center** for circles and ellipses. - The **vertex** and **focus** for parabolas. - **Vertices**, **foci**, and **asymptotes** for hyperbolas. Understanding these attributes helps in interpreting the conic鈥檚 real-world representation, such as how a parabola might model the trajectory of a thrown ball.
Conversion of Coordinates
Converting between polar and rectangular coordinates is a critical skill when dealing with conic sections. This transformation facilitates the interpretation and analysis of equations in different contexts.The key formulas for conversion are:
  • From polar to rectangular: \(x = r \cos \theta\) and \(y = r \sin \theta\).
  • From rectangular to polar: \(r = \sqrt{x^2 + y^2}\) and \(\theta = \tan^{-1}(\frac{y}{x})\).
These conversions allow us to change the viewpoint of a problem, making it easier to work with. For example, an equation in polar form that's challenging to interpret might become more understandable once translated into rectangular coordinates. This interchange between systems is particularly useful in calculus and physics, where different formulations offer different insights into a problem.

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 parametric equations for the position of an object are given. Find the object's velocity and speed at the given times and describe its motion $$\left\\{\begin{array}{ll} x=40 t+5 & \text { (a) } t=0, \text { (b) } t=2 \\ y=20+3 t-16 t^{2} & \text { . } \end{array}\right.$$

The problem of finding the slope of \(r=\sin 3 \theta\) at the point (0,0) is not a well-defined problem. To see what we mean, show that the curve passes through the origin at \(\theta=0, \theta=\frac{\pi}{3}\) and \(\theta=\frac{2 \pi}{3},\) and find the slopes at these angles. Briefly explain why they are different even though the point is the same.

In exercises find the slopes of the tangent lines to the given curves at the indicated points. $$\left\\{\begin{array}{ll} x=\cos 2 t & \text { (a) } t=\frac{\pi}{2}, \text { (b) } t=\frac{3 \pi}{2}, \text { (c) }(1,0) \\ y=\sin 3 t & \end{array}\right.$$

If the polar curve \(r=f(\theta), a \leq \theta \leq b,\) has length \(L,\) show that \(r=c f(\theta), a \leq \theta \leq b,\) has length \(|c| L\) for any constant \(c\).

Find parametric equations describing the given curve. The portion of the parabola \(y=2-x^{2}\) from (2,-2) to (0,2)
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

Read Explanation

Applied Mathematics

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.