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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 12.4.26
Chapter 12, Problem 12.4.26

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

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Identify the type of conic section by comparing the given equation to standard forms. The equation is \(x^{2} + \frac{y^{2}}{9} = 1\). Since both \(x^{2}\) and \(y^{2}\) terms are positive and the equation equals 1, this suggests it is an ellipse.
Rewrite the equation in the standard form of an ellipse: \(\frac{x^{2}}{1} + \frac{y^{2}}{9} = 1\). Here, \(a^{2} = 9\) and \(b^{2} = 1\), where \(a^{2}\) is the larger denominator corresponding to the major axis.
Determine the lengths of the major and minor axes. The major axis length is \(2a = 2 \times 3 = 6\), and the minor axis length is \(2b = 2 \times 1 = 2\).
Find the coordinates of the vertices. Since \(a^{2} = 9\) is under \(y^{2}\), the major axis is vertical. The vertices are at \((0, \pm a) = (0, \pm 3)\).
Calculate the foci using \(c^{2} = a^{2} - b^{2}\). Here, \(c^{2} = 9 - 1 = 8\), so \(c = \sqrt{8} = 2\sqrt{2}\). The foci are located at \((0, \pm c) = (0, \pm 2\sqrt{2})\).

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Key Concepts

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

Identification of Conic Sections

Conic sections are curves obtained by intersecting a plane with a cone, resulting in parabolas, ellipses, or hyperbolas. The general form of their equations helps classify them: ellipses have sums of squared terms equal to 1 with positive coefficients, hyperbolas have a difference of squared terms, and parabolas have one squared term. Recognizing these forms is essential to determine the type of conic.
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Properties and Features of Ellipses

An ellipse is defined by two foci such that the sum of distances from any point on the ellipse to the foci is constant. Key features include vertices (endpoints of the major axis), foci, and the lengths of the major and minor axes. Understanding these properties allows for accurate graphing and labeling of the ellipse.
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Graphing and Analyzing Conic Sections

Graphing conic sections involves plotting key points like vertices and foci, and drawing axes or asymptotes where applicable. For ellipses, labeling the major and minor axes is crucial, while hyperbolas require asymptotes. This process helps visualize the curve and understand its geometric behavior.
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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

Subtle symmetry Without using a graphing utility, determine the symmetries (if any) of the curve r=4-sin (θ/2)

Textbook Question

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


y = 3

Textbook Question

93–94. Parametric equations of ellipses Find parametric equations (not unique) of the following ellipses (see Exercises 91–92). Graph the ellipse and find a description in terms of x and y.


An ellipse centered at (-2, -3) with major and minor axes of lengths 30 and 20, parallel to the x- and y-axes, respectively, generated counterclockwise (Hint: Shift the parametric equations.)

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Textbook Question

31–38. Equations of parabolas Find an equation of the following parabolas. Unless otherwise specified, assume the vertex is at the origin.

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Textbook Question

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


r = 6 cos θ + 8 sin θ