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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 12.2.43
Chapter 12, Problem 12.2.43

37–48. Polar-to-Cartesian coordinates Convert the following equations to Cartesian coordinates. Describe the resulting curve.


r = 6 cos θ + 8 sin θ

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Recall the relationships between polar and Cartesian coordinates: \(x = r \cos \theta\) and \(y = r \sin \theta\). Also, \(r^2 = x^2 + y^2\).
Start with the given polar equation: \(r = 6 \cos \theta + 8 \sin \theta\).
Multiply both sides of the equation by \(r\) to eliminate the denominator when substituting: \(r \cdot r = r (6 \cos \theta + 8 \sin \theta)\), which gives \(r^2 = 6r \cos \theta + 8r \sin \theta\).
Substitute \(r^2\) with \(x^2 + y^2\), \(r \cos \theta\) with \(x\), and \(r \sin \theta\) with \(y\) to get: \(x^2 + y^2 = 6x + 8y\).
Rewrite the equation to standard form by bringing all terms to one side: \(x^2 - 6x + y^2 - 8y = 0\). Then, complete the square for both \(x\) and \(y\) terms to identify the type and properties of the curve.

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Key Concepts

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

Polar and Cartesian Coordinate Systems

Polar coordinates represent points using a radius and an angle (r, θ), while Cartesian coordinates use (x, y) positions on a plane. Understanding how to convert between these systems is essential for analyzing curves defined in polar form.
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Intro to Polar Coordinates

Conversion Formulas Between Polar and Cartesian Coordinates

The key formulas for conversion are x = r cos θ and y = r sin θ. Additionally, r² = x² + y² and tan θ = y/x. These relationships allow rewriting polar equations in terms of x and y to identify the curve in Cartesian form.
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Intro to Polar Coordinates

Identifying and Describing Curves from Equations

After converting to Cartesian form, recognizing the type of curve (e.g., circle, ellipse, line) involves rearranging the equation and comparing it to standard forms. This helps in describing the geometric shape represented by the original polar equation.
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Related Practice
Textbook Question

31–36. Converting coordinates Express the following Cartesian coordinates in polar coordinates in at least two different ways.


(-4, 4√3)

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Textbook Question

49–52. Cartesian-to-polar coordinates Convert the following equations to polar coordinates.


y = 3

Textbook Question

81–88. Arc length Find the arc length of the following curves on the given interval.


x = 3 cos t, y = 3 sin t + 1; 0 ≤ t ≤ 2π

Textbook Question

37–52. Curves to parametric equations Find parametric equations for the following curves. Include an interval for the parameter values. Answers are not unique.


The line that passes through the points P(1, 1) and Q(3, 5), oriented in the direction of increasing x

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Textbook Question

81–88. Arc length Find the arc length of the following curves on the given interval.


x = eᵗ sin t, y = eᵗ cos t; 0 ≤ t ≤ 2π

Textbook Question

13–30. Graphing conic sections Determine whether the following equations describe a parabola, an ellipse, or a hyperbola, and then sketch a graph of the curve. For each parabola, specify the location of the focus and the equation of the directrix; for each ellipse, label the coordinates of the vertices and foci, and find the lengths of the major and minor axes; for each hyperbola, label the coordinates of the vertices and foci, and find the equations of the asymptotes.


x² + y²/9 = 1