Domanda del libro di testo
25–30. Converting coordinates Express the following polar coordinates in Cartesian coordinates.
(2, 7π/4)
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Ch.12 - Parametric and Polar Curves
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 12, Problema 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
Guida verificata passo dopo passo1
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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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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Percorso guidato
5:33
Parabolas as Conic Sections
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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Percorso guidato
06:21Properties of Functions
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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Percorso guidato
5:33Parabolas as Conic Sections
Pratica correlata
Domanda del libro di testo
33–40. Areas of regions Make a sketch of the region and its bounding curves. Find the area of the region.
The region inside the limaçon r = 2 + cos θ
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Domanda del libro di testo
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 = 3 cos t, y = 3 sin t; π ≤ t ≤ 2π
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Domanda del libro di testo
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 = 3 + t, y = 1 − t; 0 ≤ t ≤ 1
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Domanda del libro di testo
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.
4x = -y²
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Domanda del libro di testo
90–94. Focal chords A focal chord of a conic section is a line through a focus joining two points of the curve. The latus rectum is the focal chord perpendicular to the major axis of the conic. Prove the following properties.
The length of the latus rectum of the parabola y ² =4px or x ² =4py is 4|p|.
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