Ch.12 - Parametric and Polar Curves

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch.12 - Parametric and Polar Curves

Problem 12.1.94

Chapter 12, Problem 12.1.94

93–94. Parametric equations of ellipses Find parametric equations (not unique) of the following ellipses (see Exercises 91–92). Graph the ellipse and find a description in terms of x and y.

An ellipse centered at (-2, -3) with major and minor axes of lengths 30 and 20, parallel to the x- and y-axes, respectively, generated counterclockwise (Hint: Shift the parametric equations.)

Verified step by step guidance

Identify the center of the ellipse as \((-2, -3)\), the lengths of the major and minor axes as 30 and 20, respectively, and note that the axes are parallel to the x- and y-axes.

Recall the standard parametric equations for an ellipse centered at the origin with semi-major axis \(a\) and semi-minor axis \(b\): \[x = a \cos(t), \quad y = b \sin(t)\] where \(t\) is the parameter varying from \(0\) to \(2\pi\).

Calculate the semi-major axis \(a\) and semi-minor axis \(b\) by dividing the lengths of the axes by 2: \[a = \frac{30}{2} = 15, \quad b = \frac{20}{2} = 10\]

Shift the parametric equations to account for the center \((-2, -3)\) by adding the center coordinates to the \(x\) and \(y\) components: \[x = -2 + 15 \cos(t), \quad y = -3 + 10 \sin(t)\]

Write the equation of the ellipse in terms of \(x\) and \(y\) by using the standard form centered at \((-2, -3)\): \[\frac{(x + 2)^2}{15^2} + \frac{(y + 3)^2}{10^2} = 1\]

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

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

Parametric Equations of an Ellipse

Parametric equations express the coordinates of points on an ellipse as functions of a parameter, usually t. For an ellipse centered at the origin with axes aligned to the coordinate axes, the standard form is x = a cos(t), y = b sin(t), where a and b are the semi-major and semi-minor axes. These equations trace the ellipse as t varies from 0 to 2π.

Shifting the Center of an Ellipse

When an ellipse is not centered at the origin, its parametric equations must be adjusted by adding the center coordinates to the standard form. If the center is at (h, k), the equations become x = h + a cos(t) and y = k + b sin(t). This shift translates the ellipse without changing its shape or orientation.

Relationship Between Parametric and Cartesian Forms

The Cartesian equation of an ellipse relates x and y directly, typically in the form ((x - h)^2)/a^2 + ((y - k)^2)/b^2 = 1. Parametric equations provide a way to generate points on the ellipse, while the Cartesian form describes all points satisfying the ellipse's geometric definition. Understanding both forms helps in graphing and analyzing ellipses.

Recommended video:

Guided course

06:03

Parameterizing Equations of Circles & Ellipses

Related Practice

Textbook Question

Subtle symmetry Without using a graphing utility, determine the symmetries (if any) of the curve r=4-sin (θ/2)

Textbook Question

49–52. Cartesian-to-polar coordinates Convert the following equations to polar coordinates.

y = 3

Textbook Question

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

views

Textbook Question

37–52. Curves to parametric equations Find parametric equations for the following curves. Include an interval for the parameter values. Answers are not unique.

The line segment starting at P(0, 0) and ending at Q(2, 8)

views

Textbook Question

37–52. Curves to parametric equations Find parametric equations for the following curves. Include an interval for the parameter values. Answers are not unique.

The horizontal line segment starting at P(8, 2) and ending at Q(−2, 2)

views

Textbook Question

13–30. Graphing conic sections Determine whether the following equations describe a parabola, an ellipse, or a hyperbola, and then sketch a graph of the curve. For each parabola, specify the location of the focus and the equation of the directrix; for each ellipse, label the coordinates of the vertices and foci, and find the lengths of the major and minor axes; for each hyperbola, label the coordinates of the vertices and foci, and find the equations of the asymptotes.

x² + y²/9 = 1