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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.54a
Chapter 12, Problem 12.R.54a

53–57. Conic sections
a. Determine whether the following equations describe a parabola, an ellipse, or a hyperbola.
x = 16y²

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1
Recall the general forms of conic sections: - Parabola: one variable is squared, the other is to the first power (e.g., \(y^2 = 4ax\) or \(x^2 = 4ay\)). - Ellipse: both variables are squared with the same sign and different coefficients (e.g., \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\)). - Hyperbola: both variables are squared but with opposite signs (e.g., \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) or \(-\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\)).
Look at the given equation: \(x = 16y^2\). Notice that \(x\) is to the first power and \(y\) is squared.
Rewrite the equation to isolate zero on one side: \(x - 16y^2 = 0\). This shows only one variable is squared and the other is linear.
Since only one variable is squared and the other is linear, this matches the form of a parabola, where the squared term is on one side and the linear term on the other.
Therefore, conclude that the equation \(x = 16y^2\) represents a parabola.

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Key Concepts

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

Conic Sections and Their Standard Forms

Conic sections are curves obtained by intersecting a plane with a double-napped cone, resulting in parabolas, ellipses, or hyperbolas. Each conic has a standard equation form, such as y² = 4ax for parabolas, (x²/a²) + (y²/b²) = 1 for ellipses, and (x²/a²) - (y²/b²) = 1 for hyperbolas. Recognizing these forms helps classify the given equation.
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Identifying a Parabola from Its Equation

A parabola is characterized by an equation where one variable is squared and the other is to the first power, typically in the form x = ay² or y = ax². The given equation x = 16y² fits this pattern, indicating it is a parabola opening along the x-axis. Understanding this helps distinguish it from ellipses and hyperbolas.
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Graphical Interpretation of Conic Sections

Graphing the equation provides visual insight into the conic's shape and orientation. For x = 16y², the parabola opens rightward since x depends on y² with a positive coefficient. Recognizing the direction and shape of the graph aids in confirming the type of conic section represented by the equation.
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Related Practice
Textbook Question

51–52. {Use of Tech} Arc length of polar curves Find the approximate length of the following curves.

The limaçon r=3−6cosθ

Textbook Question

Polar conversion Write the equation r ² +r(2sinθ−6cosθ)=0 in Cartesian coordinates and identify the corresponding curve.

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Textbook Question

53–57. Conic sections

c. Find the eccentricity of the curve.

y² - 4x² = 16

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Textbook Question

A polar conic section Consider the equation r² = sec2θ

b. Find the vertices, foci, directrices, and eccentricity of the curve."

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Textbook Question

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


a. The point with Cartesian coordinates (−2, 2) has polar coordinates (2√2, 3π/4), (2√2, 11π/4), (2√2, −5π/4), and (−2√2,−π/4).  

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Textbook Question

67–72. Derivatives Consider the following parametric curves.

a. Determine dy/dx in terms of t and evaluate it at the given value of t.


x = t + 1/t, y = t − 1/t; t = 1

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