Ch.12 - Parametric and Polar Curves

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch.12 - Parametric and Polar Curves

Problem 12.4.12

Chapter 12, Problem 12.4.12

How does the eccentricity determine the type of conic section?

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Recall that the eccentricity \(e\) of a conic section is a non-negative real number that measures how much the conic deviates from being circular.

Understand that the eccentricity is defined as the ratio of the distance from any point on the conic to the focus, divided by the perpendicular distance from that point to the directrix.

Use the value of \(e\) to classify the conic: if \(e = 0\), the conic is a circle; if \(0 < e < 1\), it is an ellipse; if \(e = 1\), it is a parabola; and if \(e > 1\), it is a hyperbola.

Recognize that this classification arises because the shape of the conic changes as the eccentricity changes, reflecting how 'stretched' or 'open' the curve is.

Summarize that eccentricity provides a precise numerical way to distinguish between different conic sections based on their geometric properties.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Eccentricity of a Conic Section

Eccentricity (denoted as e) is a non-negative real number that measures the deviation of a conic section from being circular. It is defined as the ratio of the distance from any point on the conic to the focus, over the perpendicular distance to the directrix. This value uniquely characterizes the shape of the conic.

Classification of Conic Sections by Eccentricity

The type of conic section is determined by the value of eccentricity: if e = 0, the conic is a circle; if 0 < e < 1, it is an ellipse; if e = 1, it is a parabola; and if e > 1, it is a hyperbola. This classification helps in identifying the conic based on geometric properties.

Geometric Definition of Conics Using Focus and Directrix

Conic sections can be defined as the set of points where the ratio of distances to a fixed point (focus) and a fixed line (directrix) is constant and equal to eccentricity. Understanding this geometric definition is essential to grasp how eccentricity influences the shape and type of the conic.

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Parabolas as Conic Sections

Related Practice

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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.

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25–30. Converting coordinates Express the following polar coordinates in Cartesian coordinates.

(4, 5π)

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Textbook Question

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