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Ch.12 - Parametric and Polar Curves
Briggs - Calculus: Early Transcendentals 3rd Edition
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Tutti i libri di testoBriggs 3rd EditionCh.12 - Parametric and Polar CurvesProblema 12.R.70
Capitolo 12, Problema 12.R.70

Conic parameters: A hyperbola has eccentricity e = 2 and foci (0, ±2). Find the location of the vertices and directrices.

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Identify the given parameters: eccentricity \(e = 2\) and foci at \((0, \pm 2)\). Since the foci lie on the y-axis, the hyperbola is vertical with center at the origin \((0,0)\).
Recall the relationship between the center, foci, and vertices for a vertical hyperbola: the foci are at \((0, \pm c)\) and the vertices are at \((0, \pm a)\), where \(c\) is the focal distance and \(a\) is the distance from the center to each vertex.
From the foci coordinates, determine \(c\): since the foci are at \((0, \pm 2)\), we have \(c = 2\).
Use the eccentricity formula for a hyperbola: \(e = \frac{c}{a}\). Given \(e = 2\) and \(c = 2\), solve for \(a\) by rearranging to \(a = \frac{c}{e}\).
Find the directrices using the formula for vertical hyperbolas: the directrices are horizontal lines given by \(y = \pm \frac{a}{e}\). Substitute the values of \(a\) and \(e\) to find their locations.

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Eccentricity of a Hyperbola

Eccentricity (e) measures how much a conic section deviates from being circular. For a hyperbola, e > 1, and it relates the distances between the foci and vertices. It is defined as e = c/a, where c is the distance from the center to a focus, and a is the distance from the center to a vertex.
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Asymptotes of Hyperbolas

Standard Form and Parameters of a Hyperbola

A hyperbola centered at the origin with vertical transverse axis has foci at (0, ±c) and vertices at (0, ±a). The relationship c² = a² + b² holds, where b relates to the conjugate axis. Knowing c and e allows calculation of a and b, which determine the shape and key points of the hyperbola.
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Circles in Standard Form Example 1

Directrices of a Hyperbola

Directrices are fixed lines used to define a hyperbola via eccentricity. For a hyperbola with vertical transverse axis, the directrices are horizontal lines located at y = ±(a/e). They help in understanding the geometric definition and are essential for graphing and analyzing the hyperbola.
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Asymptotes of Hyperbolas
Pratica correlata
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44–49. Areas of regions Find the area of the following regions.

The region inside the cardioid r=1+cosθ and outside the cardioid r=1−cosθ

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7–8. Parametric curves and tangent lines

a. Eliminate the parameter to obtain an equation in x and y.

x = 4sin 2t, y = 3cos 2t, for 0 ≤ t ≤ π; t = π/6

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10–12. Parametric curves

a. Eliminate the parameter to obtain an equation in x and y.

x = ln t, y = 8ln t², for 1 ≤ t ≤ e²; (1, 16)

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53–57. Conic sections

b. Use analytical methods to determine the location of the foci, vertices, and directrices.

x² - y²/2 = 1

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10–12. Parametric curves

a. Eliminate the parameter to obtain an equation in x and y.

x = t² + 4, y = -t, for -2 < t < 0; (5, 1)

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10–12. Parametric curves

a. Eliminate the parameter to obtain an equation in x and y.

x = 3cos(-t), y = 3sin(-t) - 1, for 0 ≤ t ≤ π; (0, -4)

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