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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 11.7.12
Chapter 11, Problem 11.7.12

Ellipses and Eccentricity


Exercises 9–12 give the foci or vertices and the eccentricities of ellipses centered at the origin of the xy-plane. In each case, find the ellipse’s standard-form equation in Cartesian coordinates.


Vertices: (±10,0)
Eccentricity: 0.24

Verified step by step guidance
1
Identify the orientation of the ellipse based on the vertices. Since the vertices are at (±10, 0), the major axis is along the x-axis, so the ellipse equation will be of the form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\).
Determine the value of \(a\), which is the distance from the center to each vertex. Here, \(a = 10\), so \(a^2 = 100\).
Recall the relationship between the eccentricity \(e\), the semi-major axis \(a\), and the distance to each focus \(c\): \(e = \frac{c}{a}\). Use the given eccentricity \(e = 0.24\) to find \(c = e \times a\).
Use the relationship between \(a\), \(b\), and \(c\) for ellipses: \(c^2 = a^2 - b^2\). Rearrange this to solve for \(b^2\): \(b^2 = a^2 - c^2\).
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 ellipse.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Standard Form of an Ellipse

The standard form of an ellipse centered at the origin is either (x²/a²) + (y²/b²) = 1 for a horizontal major axis or (x²/b²) + (y²/a²) = 1 for a vertical major axis, where a is the semi-major axis length and b is the semi-minor axis length. Knowing the vertices helps determine the value of a.
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Circles in Standard Form Example 1

Eccentricity of an Ellipse

Eccentricity (e) measures how elongated an ellipse is, defined as e = c/a, where c is the distance from the center to each focus and a is the semi-major axis length. It ranges from 0 (circle) to 1 (parabola). Given e and a, you can find c, which helps locate the foci.
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Foci and Vertices of an Ellipse

Relationship Between a, b, and c in an Ellipse

In an ellipse, the relationship c² = a² - b² connects the semi-major axis (a), semi-minor axis (b), and focal distance (c). This formula allows you to find b once a and c are known, completing the parameters needed to write the ellipse's equation.
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Finding Area Between Curves that Cross on the Interval Example 3
Related Practice
Textbook Question

Finding Cartesian from Parametric Equations


In Exercises 19–24, match the parametric equations with the parametric curves labeled A through F.


x = cos t, y = sin 3t


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Textbook Question

Finding Lengths of Polar Curves


Find the lengths of the curves in Exercises 21–28.


The curve r = cos³(θ/3), 0 ≤ θ ≤ π/4

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Textbook Question

Identifying Graphs


Match the parabolas in Exercises 1−4 with the following equations: x² = 2y, x² = −6y, y² = 8x, y² = −4x


Then find each parabola's focus and directrix.



Textbook Question

Surface Area


Find the areas of the surfaces generated by revolving the curves in Exercises 31-34 about the indicated axes.


x = t + √2, y = (t²/2) + √2t, −√2 ≤ t ≤ √2; y−axis

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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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