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Ch.12 - Parametric and Polar Curves
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh.12 - Parametric and Polar CurvesProblem 12.4.43
Chapter 12, Problem 12.4.43

39–50. Equations of ellipses and hyperbolas Find an equation of the following ellipses and hyperbolas, assuming the center is at the origin. 
An ellipse with vertices (±5, 0), passing through the point (4, 3/5)

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Identify the standard form of the ellipse equation centered at the origin with a horizontal major axis: \(\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1\).
From the vertices given as \((\pm 5, 0)\), determine the value of \(a\) since the vertices lie along the major axis. Here, \(a = 5\), so \(a^{2} = 25\).
Substitute \(a^{2} = 25\) into the ellipse equation to get \(\frac{x^{2}}{25} + \frac{y^{2}}{b^{2}} = 1\).
Use the point \((4, \frac{3}{5})\) that lies on the ellipse to substitute \(x = 4\) and \(y = \frac{3}{5}\) into the equation, resulting in \(\frac{4^{2}}{25} + \frac{(\frac{3}{5})^{2}}{b^{2}} = 1\).
Solve the resulting equation for \(b^{2}\) to find the value of \(b^{2}\), which completes 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 Equation

An ellipse centered at the origin with a horizontal major axis has the equation (x²/a²) + (y²/b²) = 1, where 'a' is the distance from the center to each vertex along the x-axis, and 'b' is the distance along the y-axis. Knowing the vertices helps determine 'a', which is essential for writing the ellipse equation.
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Parameterizing Equations of Circles & Ellipses

Using a Point on the Ellipse to Find Parameters

Substituting a known point on the ellipse into the standard equation allows solving for the unknown parameter 'b'. This step ensures the ellipse passes through the given point, refining the equation to fit the specific ellipse described.
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Relationship Between 'a', 'b', and the Shape of the Ellipse

The values of 'a' and 'b' determine the ellipse's shape and size. 'a' is the semi-major axis length, and 'b' is the semi-minor axis length. Understanding their relationship helps in interpreting the ellipse's geometry and verifying the correctness of the equation.
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Related Practice
Textbook Question

31–36. Eliminating the parameter Eliminate the parameter to express the following parametric equations as a single equation in x and y.


x=2 sin 8t, y=2 cos 8t 

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 = 12y

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.


A circle centered at the origin with radius 4, generated counterclockwise

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Textbook Question

15–30. Working with parametric equations Consider the following parametric equations.

a. Eliminate the parameter to obtain an equation in x and y.

b. Describe the curve and indicate the positive orientation.


x = cos t, y = sin² t; 0 ≤ t ≤ π

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Textbook Question

63–74. Arc length of polar curves Find the length of the following polar curves.


{Use of Tech} The complete limaçon r=4−2cosθ

Textbook Question

11–20. Slopes of tangent lines Find the slope of the line tangent to the following polar curves at the given points.


r = 1 - sin θ; (1/2, π/6)

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