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Calculus / Calculus: Early Transcendentals 1 / Chapter 10.1 / Problem 59E

Calculus: Early Transcendentals | 1st Edition | ISBN: 9780321570567 | Authors: William L. Briggs, Lyle Cochran, Bernard Gillett

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Parametric equations of ellipses Find parametric equations

Chapter 8, Problem 59E

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QUESTION:

Find parametric equations of the following ellipses (see Exercises 57-58). Graph the ellipse and find a description in terms of x and y. Solutions are not unique.

An ellipse centered at the origin with major axis of length 6 on the x-axis and minor axis of length 3 on the y-axis, generated counterclockwise

Questions & Answers

QUESTION:

Find parametric equations of the following ellipses (see Exercises 57-58). Graph the ellipse and find a description in terms of x and y. Solutions are not unique.

An ellipse centered at the origin with major axis of length 6 on the x-axis and minor axis of length 3 on the y-axis, generated counterclockwise

ANSWER:

Solution 59E

Step 1:

An ellipse is generated by the parametric equations $x=a\ cos(t)$ , $y=b\ sin(t)$ . If 0<a<b, then the long axis (or major axis) lies on the y-axis and the short axis (or minor axis) lies on the x-axis. If 0<b<a, the axes are reversed. The lengths of the axes in the x- and y-directions are 2a and 2b, respectively.

The general equation of an ellipse is given by

$\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1$

Where $a\$ is the radius of the semi-major axis and $b$ is the radius of the semi minor axis.

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