Domanda del libro di testo
Write the equations that are used to express a point with polar coordinates (r, θ) in Cartesian coordinates.
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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 12.4.54
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
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Identify the orientation of the ellipse based on the vertices. Since the vertices are at (0, ±9), the major axis is vertical, so the ellipse is vertical with center at the origin.
Recall the standard form of the ellipse equation with a vertical major axis: \(\frac{x^{2}}{b^{2}} + \frac{y^{2}}{a^{2}} = 1\), where \(a\) is the semi-major axis length and \(b\) is the semi-minor axis length, with \(a > b\).
Determine the value of \(a\) from the vertices. Since the vertices are at (0, ±9), the distance from the center to each vertex is \(a = 9\).
Use the eccentricity formula for an ellipse: \(e = \frac{c}{a}\), where \(c\) is the distance from the center to each focus. Given \(e = \frac{1}{4}\), solve for \(c\) as \(c = e \times a\).
Find \(b\) using the relationship \(c^{2} = a^{2} - b^{2}\). Rearrange to solve for \(b^{2} = a^{2} - c^{2}\). Then write the equation of the ellipse using the values of \(a\) and \(b\).
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Ellipse Definition and Properties
An ellipse is the set of points where the sum of distances to two fixed points (foci) is constant. Key features include vertices, foci, center, and axes lengths. The vertices lie on the major axis, and the distance between the center and vertices defines the semi-major axis length.
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Percorso guidato
06:21
Properties of Functions
Eccentricity and Directrix of an Ellipse
Eccentricity (e) measures how 'stretched' an ellipse is, defined as the ratio of the distance from a focus to a point on the ellipse over the distance from that point to the corresponding directrix. For ellipses, e is between 0 and 1. The directrix is a fixed line used with eccentricity to define the ellipse geometrically.
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5:30Foci and Vertices of an Ellipse
Equation of an Ellipse Centered at the Origin
The standard form of an ellipse centered at the origin with vertical major axis is (x²/b²) + (y²/a²) = 1, where a > b. The vertices are at (0, ±a), and the foci at (0, ±c), with c² = a² - b². Using eccentricity e = c/a helps find c and b, enabling the full equation.
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4:50Graph Ellipses NOT at Origin
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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.
25y² - 4x² = 100
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Domanda del libro di testo
What is the polar equation of the vertical line x = 5?
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Domanda del libro di testo
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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Domanda del libro di testo
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 segment starting at P(0, 0) and ending at Q(2, 8)
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Domanda del libro di testo
What is the slope of the line θ=π/3?
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