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Finding Polar Areas
Find the areas of the regions in Exercises 9–18.
Shared by the circles r = 1 and r = 2 sin θ
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Ch. 11 - Parametric Equations and Polar Coordinates
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
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Hass 15th EditionCh. 11 - Parametric Equations and Polar CoordinatesProblema 11.7.20
Hass 15th EditionCh. 11 - Parametric Equations and Polar CoordinatesProblema 11.7.20Capitolo 11, Problema 11.7.20
Hyperbolas and Eccentricity
In Exercises 17-24, find the eccentricity of the hyperbola. Then find and graph the hyperbola's foci and directrices.
y² − x² = 4
Guida verificata passo dopo passo1
Rewrite the given hyperbola equation \(y^{2} - x^{2} = 4\) in the standard form by dividing both sides by 4, resulting in \(\frac{y^{2}}{4} - \frac{x^{2}}{4} = 1\).
Identify the values of \(a^{2}\) and \(b^{2}\) from the standard form. Here, \(a^{2} = 4\) and \(b^{2} = 4\). Since the \(y^{2}\) term is positive and comes first, the hyperbola opens vertically.
Calculate \(c^{2}\) using the relationship for hyperbolas: \(c^{2} = a^{2} + b^{2}\). Substitute the values to get \(c^{2} = 4 + 4\).
Find the eccentricity \(e\) using the formula \(e = \frac{c}{a}\). Take the square root of \(c^{2}\) to find \(c\), then divide by \(a\) (which is the square root of \(a^{2}\)).
Determine the coordinates of the foci, which lie along the transverse axis (vertical axis here) at \((0, \pm c)\), and write the equations of the directrices, which are horizontal lines given by \(y = \pm \frac{a^{2}}{c}\).
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Standard Form of a Hyperbola
A hyperbola can be expressed in the standard form (y²/a²) - (x²/b²) = 1 or (x²/a²) - (y²/b²) = 1. This form helps identify the values of a² and b², which are essential for determining the hyperbola's shape, orientation, and key features like vertices and asymptotes.
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Percorso guidato
3:40
Circles in Standard Form Example 1
Eccentricity of a Hyperbola
Eccentricity (e) measures how 'stretched' a hyperbola is and is defined as e = c/a, where c² = a² + b². It is always greater than 1 for hyperbolas, indicating the distance of the foci from the center relative to the vertices.
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Percorso guidato
5:50Asymptotes of Hyperbolas
Foci and Directrices of a Hyperbola
The foci are two fixed points located along the transverse axis, at a distance c from the center. The directrices are lines perpendicular to the transverse axis, located at a distance a/e from the center. Both are used to define and graph the hyperbola accurately.
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5:22Foci and Vertices of Hyperbolas
Pratica correlata
Domanda del libro di testo
Cartesian to Polar Equations
Replace the Cartesian equations in Exercises 53–66 with equivalent polar equations.
(x + 2)² + (y − 5)² = 16"
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Domanda del libro di testo
Finding Cartesian from Parametric Equations
Exercises 1–18 give parametric equations and parameter intervals for the motion of a particle in the xy-plane. Identify the particle’s path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion.
x=√(t+1), y=√t, t ≥ 0
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Domanda del libro di testo
Lines
Sketch the lines in Exercises 45–48 and find Cartesian equations for them.
r cos (θ + π/3) = 2
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Domanda del libro di testo
Hyperbolas and Eccentricity
Exercises 25–28 give the eccentricities and the vertices or foci of hyperbolas centered at the origin of the xy-plane. In each case, find the hyperbola’s standard-form equation in Cartesian coordinates.
Eccentricity: 1.25
Foci: (0, ±5)
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Polar Coordinates
Exercises 19–22 give the eccentricities of conic sections with one focus at the origin of the polar coordinate plane, along with the directrix for that focus. Find a polar equation for each conic section.
e = 1/3, r sin θ = −6
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