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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 11.7.20
Chapter 11, Problem 11.7.20

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

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1
Rewrite the given hyperbola equation \(y^{2} - x^{2} = 4\) in the standard form by dividing both sides by 4, resulting in \(\frac{y^{2}}{4} - \frac{x^{2}}{4} = 1\).
Identify the values of \(a^{2}\) and \(b^{2}\) from the standard form. Here, \(a^{2} = 4\) and \(b^{2} = 4\). Since the \(y^{2}\) term is positive and comes first, the hyperbola opens vertically.
Calculate \(c^{2}\) using the relationship for hyperbolas: \(c^{2} = a^{2} + b^{2}\). Substitute the values to get \(c^{2} = 4 + 4\).
Find the eccentricity \(e\) using the formula \(e = \frac{c}{a}\). Take the square root of \(c^{2}\) to find \(c\), then divide by \(a\) (which is the square root of \(a^{2}\)).
Determine the coordinates of the foci, which lie along the transverse axis (vertical axis here) at \((0, \pm c)\), and write the equations of the directrices, which are horizontal lines given by \(y = \pm \frac{a^{2}}{c}\).

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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

A hyperbola can be expressed in the standard form (y²/a²) - (x²/b²) = 1 or (x²/a²) - (y²/b²) = 1. This form helps identify the values of a² and b², which are essential for determining the hyperbola's shape, orientation, and key features like vertices and asymptotes.
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Circles in Standard Form Example 1

Eccentricity of a Hyperbola

Eccentricity (e) measures how 'stretched' a hyperbola is and is defined as e = c/a, where c² = a² + b². It is always greater than 1 for hyperbolas, indicating the distance of the foci from the center relative to the vertices.
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Foci and Directrices of a Hyperbola

The foci are two fixed points located along the transverse axis, at a distance c from the center. The directrices are lines perpendicular to the transverse axis, located at a distance a/e from the center. Both are used to define and graph the hyperbola accurately.
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Related Practice
Textbook Question

Finding Polar Areas


Find the areas of the regions in Exercises 9–18.


Shared by the circles r = 1 and r = 2 sin θ

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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

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

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)

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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