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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.R.17
Chapter 12, Problem 12.R.17

14–18. Parametric descriptions Write parametric equations for the following curves. Solutions are not unique.
The circle x ² + y ² =9, generated clockwise

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Recall that the standard parametric equations for a circle centered at the origin with radius \(r\) are \(x = r \cos(t)\) and \(y = r \sin(t)\), where \(t\) is the parameter representing the angle in radians.
Since the given circle is \(x^2 + y^2 = 9\), the radius \(r\) is \(3\). So the standard parametric form is \(x = 3 \cos(t)\) and \(y = 3 \sin(t)\).
Note that the standard parametric form traces the circle counterclockwise as \(t\) increases. To generate the circle clockwise, we need to reverse the direction of traversal.
To reverse the direction, replace \(t\) by \(-t\) in the parametric equations. This gives \(x = 3 \cos(-t)\) and \(y = 3 \sin(-t)\).
Use the even and odd properties of cosine and sine: \(\cos(-t) = \cos(t)\) and \(\sin(-t) = -\sin(t)\). So the parametric equations for the circle traced clockwise are \(x = 3 \cos(t)\) and \(y = -3 \sin(t)\).

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Key Concepts

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

Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of a parameter, usually denoted t. Instead of y as a function of x, both x and y depend on t, allowing more flexible descriptions of curves, including direction and speed.
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Parameterizing Equations

Equation of a Circle

The standard equation x² + y² = r² represents a circle centered at the origin with radius r. For r = 3, the circle includes all points (x, y) satisfying x² + y² = 9, forming a closed curve equidistant from the center.
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Parameterizing Equations of Circles & Ellipses

Orientation and Direction in Parametric Curves

Parametric curves have an orientation determined by how the parameter t increases. For a circle, the direction can be counterclockwise or clockwise, which affects the sign and order of the parametric functions, such as using sine and cosine with positive or negative signs.
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Related Practice
Textbook Question

65–68. Eccentricity-directrix approach Find an equation of the following curves, assuming the center is at the origin. Graph the curve, labeling vertices, foci, asymptotes (if they exist), and directrices.

A hyperbola with vertices (±4, 0) and directrices x = ±2

Textbook Question

19–20. Area bounded by parametric curves Find the area of the following regions. (Hint: See Exercises 103–105 in Section 12.1.) The region bounded by the y-axis and the parametric curve

The region bounded by the x-axis and the parametric curve x=cost, y=sin2t, for 0≤t≤π/2

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Textbook Question

58–59. Tangent lines Find an equation of the line tangent to the following curves at the given point. Check your work with a graphing utility.

x²/16 - y²/9 = 1; (20/3, -4)

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Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

e. The hyperbola y²/2 - x²/4 = 1 has no x-intercept.

Textbook Question

53–57. Conic sections

d. Make an accurate graph of the curve.

x = 16y²

Textbook Question

Polar conversion Consider the equation r=4/(sinθ+cosθ). 


a. Convert the equation to Cartesian coordinates and identify the curve it describes.  

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