Ch.12 - Parametric and Polar Curves

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch.12 - Parametric and Polar Curves

Problema 12.4.76

Capitolo 12, Problema 12.4.76

75–76. Graphs to polar equations Find a polar equation for each conic section. Assume one focus is at the origin.

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Identify the key elements from the graph: the focus is at the origin (0,0), the directrix is the vertical line \(x=2\), and the conic appears to be an ellipse.

Recall the general polar form of a conic section with one focus at the origin: \(r = \frac{ed}{1 + e \cos \theta}\), where \(e\) is the eccentricity and \(d\) is the distance from the focus to the directrix.

From the graph, determine the distance \(d\) from the origin (focus) to the directrix line \(x=2\). Since the directrix is vertical at \(x=2\), \(d=2\).

Find the eccentricity \(e\) by using the information about the ellipse. For an ellipse, \(e < 1\). Use the given points or the shape to estimate or calculate \(e\) if possible, or use the relationship between the ellipse's dimensions and \(e\).

Substitute the values of \(e\) and \(d\) into the polar equation \(r = \frac{ed}{1 + e \cos \theta}\) to write the polar equation of the conic section.

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Polar Equations of Conic Sections

Polar equations represent conic sections using a focus at the origin and a directrix line. The general form is r = ed / (1 ± e cos θ) or r = ed / (1 ± e sin θ), where e is the eccentricity and d is the distance from the focus to the directrix. This form helps describe ellipses, parabolas, and hyperbolas in polar coordinates.

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Parabolas as Conic Sections

Eccentricity and Its Role

Eccentricity (e) measures how much a conic deviates from being circular. For ellipses, 0 < e < 1; for parabolas, e = 1; and for hyperbolas, e > 1. It determines the shape and size of the conic and is essential in forming the polar equation relative to the focus and directrix.

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Classifying Differential Equations

Directrix and Focus Relationship

A conic section is defined as the set of points where the ratio of the distance to the focus and the distance to the directrix is constant (eccentricity). The directrix is a fixed line, and the focus is a fixed point; their positions determine the conic's orientation and equation in polar form.

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Parabolas as Conic Sections

Pratica correlata

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.

x²/9 + y²/4 = 1

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Express the polar equation r=f(θ) in parametric form in Cartesian coordinates, where θ is the parameter.

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31–36. Converting coordinates Express the following Cartesian coordinates in polar coordinates in at least two different ways.

(1, √3)

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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 upper half of the parabola x=y ², originating at (0, 0)

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45–60. Areas of regions Find the area of the following regions.

The region inside the lemniscate r² = 6 sin 2θ

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What is the polar equation of a circle of radius √(a²+b²) centered at (a, b)?

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