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Ch. 11 - Parametric Equations and Polar Coordinates
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 11 - Parametric Equations and Polar CoordinatesProblem 11.7.28
Chapter 11, Problem 11.7.28

Hyperbolas and Eccentricity


Exercises 25–28 give the eccentricities and the vertices or foci of hyperbolas centered at the origin of the xy-plane. In each case, find the hyperbola’s standard-form equation in Cartesian coordinates.


Eccentricity: 1.25
Foci: (0, ±5)

Verified step by step guidance
1
Identify the orientation of the hyperbola based on the foci coordinates. Since the foci are at (0, ±5), the hyperbola opens vertically along the y-axis.
Recall the standard form of a hyperbola centered at the origin with vertical transverse axis: \(\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = -1\) or equivalently \(\frac{y^{2}}{a^{2}} - \frac{x^{2}}{b^{2}} = 1\). For vertical hyperbolas, the form is \(\frac{y^{2}}{a^{2}} - \frac{x^{2}}{b^{2}} = 1\).
Use the relationship between the eccentricity \(e\), the distance to the foci \(c\), and the distance to the vertices \(a\): \(e = \frac{c}{a}\). Given \(e = 1.25\) and \(c = 5\) (from the foci coordinates), solve for \(a\) using \(a = \frac{c}{e}\).
Calculate \(b^{2}\) using the relationship for hyperbolas: \(c^{2} = a^{2} + b^{2}\). Rearrange to find \(b^{2} = c^{2} - a^{2}\).
Write the standard form equation of the hyperbola using the values of \(a^{2}\) and \(b^{2}\) found, in the form \(\frac{y^{2}}{a^{2}} - \frac{x^{2}}{b^{2}} = 1\).

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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

A hyperbola is a set of points in a plane where the difference of the distances to two fixed points called foci is constant. It has two branches and can be oriented horizontally or vertically. The standard form of its equation depends on the orientation and the location of its center.
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Definition of the Definite Integral

Eccentricity of a Hyperbola

Eccentricity (e) measures the 'flatness' of a hyperbola and is defined as the ratio of the distance from the center to a focus (c) over the distance from the center to a vertex (a), i.e., e = c/a. For hyperbolas, e > 1, and knowing e helps relate the distances c and a to write the equation.
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Asymptotes of Hyperbolas

Standard Form 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. Here, a is the distance from the center to each vertex, c is the distance to each focus, and b is found using the relationship c² = a² + b². This form allows writing the equation once a, b, and c are known.
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Related Practice
Textbook Question

Graphing Conic Sections


Exercises 63-68 give equations for conic sections and tell how many units up or down and to the right or left each curve is to be shifted. Find an equation for the new conic section, and find the new foci, vertices, centers, and asymptotes, as appropriate. If the curve is a parabola, find the new directrix as well.


x²/169 + y²/144 = 1, right 5, up 12

Textbook Question

Cartesian to Polar Equations


Replace the Cartesian equations in Exercises 53–66 with equivalent polar equations.

(x + 2)² + (y − 5)² = 16"

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Textbook Question

Finding Cartesian from Parametric Equations


Exercises 1–18 give parametric equations and parameter intervals for the motion of a particle in the xy-plane. Identify the particle’s path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion.


x=√(t+1), y=√t, t ≥ 0

Textbook Question

Hyperbolas and Eccentricity


In Exercises 17-24, find the eccentricity of the hyperbola. Then find and graph the hyperbola's foci and directrices.


y² − x² = 4

Textbook Question

Lines


Sketch the lines in Exercises 45–48 and find Cartesian equations for them.


r cos (θ + π/3) = 2

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Textbook Question

Polar Coordinates


Exercises 19–22 give the eccentricities of conic sections with one focus at the origin of the polar coordinate plane, along with the directrix for that focus. Find a polar equation for each conic section.


e = 1/3, r sin θ = −6

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