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

Textbook Question

73–76. Tangent lines Find an equation of the line tangent to the curve at the point corresponding to the given value of t.

x=t ²−1, y=t ³ +t; t=2

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Textbook Question

81–88. Arc length Find the arc length of the following curves on the given interval.

x = 2t sin t - t² cos t, y = 2t cos t + t² sin t; 0 ≤ t ≤ π

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Textbook Question

29–32. Intersection points Use algebraic methods to find as many intersection points of the following curves as possible. Use graphical methods to identify the remaining intersection points.

r = 2 cos θ and r = 1 + cos θ

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Textbook Question

84. Arc length for polar curves: Prove that the length of the curve r = f(θ) for α ≤ θ ≤ β is

L = ∫(α to β) √(f(θ)² + f'(θ)²) dθ.

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Textbook Question

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

The region inside the rose r = 4 sin 2θ and inside the circle r = 2

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Textbook Question

11–20. Slopes of tangent lines Find the slope of the line tangent to the following polar curves at the given points.

r = 2θ; (π/2, π/4)

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