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Ch. 11 - Parametric Equations and Polar Coordinates
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 11 - Parametric Equations and Polar CoordinatesProblem 11.7.68
Chapter 11, Problem 11.7.68

Examples of Polar Equations


[Technology Exercise] Graph the lines and conic sections in Exercises 65–74.


r = −2 cos θ

Verified step by step guidance
1
Recognize that the given equation is in polar form: \(r = -2 \cos \theta\). Here, \(r\) represents the distance from the origin, and \(\theta\) is the angle measured from the positive x-axis.
Recall the relationship between polar and Cartesian coordinates: \(x = r \cos \theta\) and \(y = r \sin \theta\). These will help us understand the shape of the graph.
Multiply both sides of the equation by \(r\) to rewrite it as \(r^2 = -2r \cos \theta\). Since \(r^2 = x^2 + y^2\) and \(r \cos \theta = x\), substitute these to get \(x^2 + y^2 = -2x\).
Rearrange the equation to standard form by bringing all terms to one side: \(x^2 + y^2 + 2x = 0\). Complete the square for the \(x\) terms to identify the conic section.
Complete the square: \(x^2 + 2x + 1 + y^2 = 1\), which simplifies to \((x + 1)^2 + y^2 = 1\). This represents a circle centered at \((-1, 0)\) with radius \(1\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Polar Coordinates System

The polar coordinate system represents points in a plane using a radius and an angle, denoted as (r, θ). Here, r is the distance from the origin, and θ is the angle measured from the positive x-axis. Understanding this system is essential for interpreting and graphing polar equations.
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Intro to Polar Coordinates

Graphing Polar Equations

Graphing polar equations involves plotting points based on their radius and angle values. For equations like r = -2 cos θ, the radius can be negative, which means the point is plotted in the opposite direction of the angle θ. Recognizing how to handle negative r values is key to accurate graphing.
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Introduction to Common Polar Equations

Conic Sections in Polar Form

Certain conic sections, such as circles, ellipses, parabolas, and hyperbolas, can be expressed in polar form. Equations like r = -2 cos θ often represent circles or other conics. Understanding how conic sections translate into polar equations helps in identifying and sketching their graphs.
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Parabolas as Conic Sections
Related Practice
Textbook Question

Eccentricities and Directrices


Exercises 29–36 give the eccentricities of conic sections with one focus at the origin along with the directrix corresponding to that focus. Find a polar equation for each conic section.


e = 2, x = 4

Textbook Question

Tangent Lines to Parametrized Curves


In Exercises 1−14, find an equation for the line tangent to the curve at the point defined by the given value of t. Also, find the value of d²y/dx² at this point.


x = t + eᵗ, y = 1 − eᵗ, t = 0

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Textbook Question

Lengths of Curves


Find the lengths of the curves in Exercises 25–30.


x = cos t, y = t + sin t, 0 ≤ t ≤ π

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Textbook Question

Cartesian to Polar Equations


Replace the Cartesian equations in Exercises 53–66 with equivalent polar equations.


x - y = 3

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Textbook Question

Polar to Cartesian Equations


Replace the polar equations in Exercises 27–52 with equivalent Cartesian equations. Then describe or identify the graph.


r² = 4r sin θ

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Textbook Question

Finding Parametric Equations


In Exercises 31–36, find a parametrization for the curve.


the ray (half line) with initial point (-1,2) that passes through the point (0,0)

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