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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 12.4.45
Chapter 12, Problem 12.4.45

39–50. Equations of ellipses and hyperbolas Find an equation of the following ellipses and hyperbolas, assuming the center is at the origin. 
A hyperbola with vertices (±2, 0) and asymptotes y = ±3x/2

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Identify the orientation of the hyperbola based on the vertices. Since the vertices are at (±2, 0), the hyperbola opens horizontally along the x-axis.
Write the standard form of the hyperbola equation centered at the origin with a horizontal transverse axis: \(\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1\).
Use the vertices to find \(a\). The vertices are at (±a, 0), so from (±2, 0), we have \(a = 2\), which means \(a^{2} = 4\).
Use the slopes of the asymptotes to find \(b\). For a hyperbola with a horizontal transverse axis, the asymptotes are given by \(y = \pm \frac{b}{a} x\). Given the asymptotes \(y = \pm \frac{3}{2} x\), set \(\frac{b}{a} = \frac{3}{2}\) and solve for \(b\).
Substitute the values of \(a^{2}\) and \(b^{2}\) into the standard form equation \(\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1\) to write the 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.

Standard Form of a Hyperbola Centered at the Origin

A hyperbola centered at the origin with a horizontal transverse axis has the equation \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \). Here, \(a\) is the distance from the center to each vertex along the x-axis, and \(b\) relates to the shape of the hyperbola and its asymptotes.
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Graph Hyperbolas NOT at the Origin

Vertices of a Hyperbola

Vertices are the points where the hyperbola intersects its transverse axis. For a hyperbola centered at the origin with a horizontal transverse axis, the vertices are at \((\pm a, 0)\). Knowing the vertices helps determine the value of \(a\) in the equation.
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Foci and Vertices of Hyperbolas

Asymptotes of a Hyperbola

The asymptotes of a hyperbola provide lines that the curve approaches but never touches. For a hyperbola centered at the origin with a horizontal transverse axis, the asymptotes are given by \( y = \pm \frac{b}{a} x \). Using the slopes of the asymptotes allows solving for \(b\) once \(a\) is known.
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Asymptotes of Hyperbolas
Related Practice
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37–48. Polar-to-Cartesian coordinates Convert the following equations to Cartesian coordinates. Describe the resulting curve.


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Explain why the slope of the line θ=π/2 is undefined.

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15–30. Working with parametric equations Consider the following parametric equations.

a. Eliminate the parameter to obtain an equation in x and y.

b. Describe the curve and indicate the positive orientation.


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