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Ch.12 - Parametric and Polar Curves
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh.12 - Parametric and Polar CurvesProblem 12.R.68
Chapter 12, Problem 12.R.68

65–68. Eccentricity-directrix approach Find an equation of the following curves, assuming the center is at the origin. Graph the curve, labeling vertices, foci, asymptotes (if they exist), and directrices.
A hyperbola with vertices (±4, 0) and directrices x = ±2

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Identify the type of conic: Since the problem states it is a hyperbola with vertices at (±4, 0), the transverse axis is along the x-axis, and the center is at the origin (0,0).
Recall the standard form of a hyperbola centered at the origin with a horizontal transverse axis: \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), where \(a\) is the distance from the center to each vertex.
From the vertices (±4, 0), determine \(a = 4\), so \(a^2 = 16\).
Use the eccentricity-directrix relationship for a hyperbola: The directrices are given by \(x = \pm \frac{a}{e}\), where \(e\) is the eccentricity. Given the directrices at \(x = \pm 2\), set \(2 = \frac{a}{e}\) and solve for \(e\).
Recall the relationship between \(a\), \(b\), and \(e\) for a hyperbola: \(e = \frac{c}{a}\), where \(c^2 = a^2 + b^2\). Use the value of \(e\) found to express \(b^2\) in terms of \(a^2\) and \(e\), then write the full equation of the hyperbola.

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Key Concepts

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

Definition of a Hyperbola Using Eccentricity and Directrix

A hyperbola can be defined as the set of points where the ratio of the distance to a focus and the distance to a corresponding directrix is a constant called eccentricity (e > 1). This eccentricity-directrix definition helps derive the equation of the hyperbola when the directrices and vertices are known.
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Relationship Between Vertices, Foci, and Eccentricity

Vertices are points on the hyperbola closest to the center, and foci lie along the transverse axis beyond the vertices. The distance from the center to a vertex is 'a', to a focus is 'c', and eccentricity is defined as e = c/a. Knowing vertices and directrices allows calculation of 'a', 'c', and 'e' to find the hyperbola's equation.
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Foci and Vertices of Hyperbolas

Standard Equation of a Hyperbola Centered at the Origin

For a hyperbola centered at the origin with a horizontal transverse axis, the standard form is (x²/a²) - (y²/b²) = 1. Here, 'a' is the distance from the center to each vertex, and 'b' relates to the conjugate axis. Using eccentricity and the relationship c² = a² + b², one can find 'b' and write the full equation.
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Related Practice
Textbook Question

40–41. {Use of Tech} Slopes of tangent lines

b. Find the slope of the lines tangent to the curve at the origin (when relevant).

r =3 − 6 cos θ

Textbook Question

19–20. Area bounded by parametric curves Find the area of the following regions. (Hint: See Exercises 103–105 in Section 12.1.) The region bounded by the y-axis and the parametric curve

The region bounded by the x-axis and the parametric curve x=cost, y=sin2t, for 0≤t≤π/2

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Textbook Question

14–18. Parametric descriptions Write parametric equations for the following curves. Solutions are not unique.

The circle x ² + y ² =9, generated clockwise

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Textbook Question

58–59. Tangent lines Find an equation of the line tangent to the following curves at the given point. Check your work with a graphing utility.

x²/16 - y²/9 = 1; (20/3, -4)

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Textbook Question

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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Textbook Question

Polar conversion Consider the equation r=4/(sinθ+cosθ). 


a. Convert the equation to Cartesian coordinates and identify the curve it describes.  

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