Skip to main content
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

Verified step by step guidance
1
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.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2m
Was this helpful?

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.
Recommended video:
5:59
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.
Recommended video:
5:22
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.
Recommended video:
5:50
Asymptotes of Hyperbolas
Related Practice
Textbook Question

Golden Gate Bridge Completed in 1937, San Francisco’s Golden Gate Bridge is 2.7 km long and weighs about 890,000 tons. The length of the span between the two central towers is 1280 m; the towers themselves extend 152 m above the roadway. The cables that support the deck of the bridge between the two towers hang in a parabola (see figure). Assuming the origin is midway between the towers on the deck of the bridge, find an equation that describes the cables. How long is a guy wire that hangs vertically from the cables to the roadway 500 m from the center of the bridge? 

Textbook Question

31–38. Equations of parabolas Find an equation of the following parabolas. Unless otherwise specified, assume the vertex is at the origin.

A parabola symmetric about the y-axis that passes through the point (2, -6)

Textbook Question

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


(1, 2π/3)

1
views
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.

Textbook Question

Write the equations that are used to express a point with polar coordinates (r, θ) in Cartesian coordinates.

1
views
Textbook Question

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


The region inside the curve r = √(cos θ) and inside the circle r = 1/√2 in the first quadrant

1
views