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Ch.12 - Parametric and Polar Curves
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh.12 - Parametric and Polar CurvesProblem 12.3.29
Chapter 12, Problem 12.3.29

29–32. Intersection points Use algebraic methods to find as many intersection points of the following curves as possible. Use graphical methods to identify the remaining intersection points.


r = 2 cos θ and r = 1 + cos θ

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Start by setting the two polar equations equal to each other to find the intersection points algebraically: \[2 \cos \theta = 1 + \cos \theta\].
Rearrange the equation to isolate terms: \[2 \cos \theta - \cos \theta = 1 \implies \cos \theta = 1\].
Solve for \[\theta\] where \[\cos \theta = 1\]. Recall that \[\cos \theta = 1\] at \[\theta = 0\] (and at multiples of \[2\pi\], but consider the principal values in the interval \[[0, 2\pi)\]).
Substitute \[\theta = 0\] back into either original equation to find the corresponding \[r\] value for the intersection point.
To find any additional intersection points that may not be captured by the algebraic method, use a graphical approach by plotting both curves \[r = 2 \cos \theta\] and \[r = 1 + \cos \theta\] over the interval \[\theta \in [0, 2\pi)\] and visually identify where they intersect.

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Key Concepts

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

Polar Coordinates and Equations

Polar coordinates represent points using a radius and an angle (r, θ) instead of Cartesian (x, y). Understanding how to interpret and manipulate polar equations like r = 2 cos θ is essential for finding points where two curves intersect.
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Intro to Polar Coordinates

Solving Systems of Equations

Finding intersection points involves solving the system formed by the two given equations. In this case, equate r = 2 cos θ and r = 1 + cos θ, then solve algebraically for θ and r to find common solutions representing intersection points.
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Solving Logarithmic Equations

Graphical Analysis of Polar Curves

Graphing polar curves helps visualize intersections that may be difficult to find algebraically, especially when multiple solutions or complex angles are involved. Plotting r = 2 cos θ and r = 1 + cos θ reveals additional intersection points and confirms algebraic results.
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Slope of Polar Curves
Related Practice
Textbook Question

Find the slope of the parametric curve x=−2t ³ +1, y=3t ², for −∞<t<∞, at the point corresponding to t=2. 

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Textbook Question

15–30. Working with parametric equations Consider the following parametric equations.

a. Eliminate the parameter to obtain an equation in x and y.

b. Describe the curve and indicate the positive orientation.


x = cos t, y = 1 + sin t; 0 ≤ t ≤ 2π

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Textbook Question

29–32. Intersection points Use algebraic methods to find as many intersection points of the following curves as possible. Use graphical methods to identify the remaining intersection points.


r = 1 and r = 2 sin 2θ

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Textbook Question

What is the slope of the line θ=π/3?

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Textbook Question

53–56. Eccentricity-directrix approach Find an equation of the following curves, assuming the center is at the origin. Sketch a graph labeling the vertices, foci, asymptotes (if they exist), and directrices. Use a graphing utility to check your work.


An ellipse with vertices (0, ±9) and eccentricity ¼ 

Textbook Question

25–30. Converting coordinates Express the following polar coordinates in Cartesian coordinates.


(4, 5π)

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