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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.65
Chapter 12, Problem 12.R.65

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 (0, ±2) and directrices y = ±1

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Identify the orientation of the hyperbola based on the vertices. Since the vertices are at (0, ±2), the hyperbola opens vertically along the y-axis.
Write the standard form of the hyperbola equation with vertical transverse axis centered at the origin: \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\).
Determine the value of \(a\) using the distance from the center to each vertex. Since vertices are at (0, ±2), \(a = 2\), so \(a^2 = 4\).
Use the directrix information to find the eccentricity \(e\). The directrices are given by \(y = \pm 1\), and for a hyperbola with vertical transverse axis, the directrices are at \(y = \pm \frac{a}{e}\). Set \(\frac{a}{e} = 1\) and solve for \(e\).
Calculate \(b^2\) using the relationship \(b^2 = a^2(e^2 - 1)\). Once \(a^2\) and \(e\) are known, substitute to find \(b^2\), 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 greater than 1, called the eccentricity (e). 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. The distance from the center to a vertex is 'a', and to a focus is 'c'. The eccentricity e = c/a relates these distances and is crucial for finding the foci and writing the hyperbola's equation.
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Foci and Vertices of Hyperbolas

Equation of a Hyperbola Centered at the Origin

For a hyperbola centered at the origin with vertical transverse axis, the standard form is (y²/a²) - (x²/b²) = 1. Knowing 'a' from vertices and using eccentricity to find 'c' and 'b' allows writing the equation. Directrices help determine eccentricity and complete the equation.
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Related Practice
Textbook Question

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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Textbook Question

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


(2, 7π/4)

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Textbook Question

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


The region inside the lemniscate r² = 6 sin 2θ

Textbook Question

53–57. Conic sections

c. Find the eccentricity of the curve.

x²/4 + y²/25 = 1

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Textbook Question

3–6. Eliminating the parameter Eliminate the parameter to find a description of the following curves in terms of x and y. Give a geometric description and the positive orientation of the curve.

x = sin t - 3, y = cos t + 6; 0 ≤ t ≤ π

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Textbook Question

27–32. Polar curves Graph the following equations.


r = 3 sin 4θ

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