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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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1
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
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