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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Ch.12 - Parametric and Polar Curves
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 12, Problem 12.R.56a
53–57. Conic sections
a. Determine whether the following equations describe a parabola, an ellipse, or a hyperbola.
x²/4 + y²/25 = 1
Verified step by step guidance1
Identify the general form of the conic section equation. The given equation is \( \frac{x^2}{4} + \frac{y^2}{25} = 1 \). This resembles the standard form of conic sections centered at the origin.
Recall the standard forms:
- Ellipse: \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \)
- Hyperbola: \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) or \( -\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \)
- Parabola: Usually involves only one squared term, e.g., \( y^2 = 4ax \) or \( x^2 = 4ay \).
Compare the given equation to these forms. Since both \( x^2 \) and \( y^2 \) terms are positive and added together, and the equation equals 1, it matches the form of an ellipse.
Note the denominators: \(4\) and \(25\) represent \(a^2\) and \(b^2\) respectively, which are positive constants, confirming the ellipse shape.
Conclude that the equation \( \frac{x^2}{4} + \frac{y^2}{25} = 1 \) describes an ellipse.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Conic Sections
Conic sections are curves obtained by intersecting a plane with a double-napped cone. The main types are parabolas, ellipses, and hyperbolas, each defined by specific algebraic equations and geometric properties.
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Parabolas as Conic Sections
Standard Form of Conic Equations
Each conic section has a standard equation form: ellipses and hyperbolas involve sums or differences of squared terms set equal to 1, while parabolas have a single squared term. Recognizing these forms helps identify the conic type.
Recommended video:
5:18Circles in Standard Form
Ellipse Identification Criteria
An ellipse equation typically has the form (x²/a²) + (y²/b²) = 1 with both denominators positive and the sum of squared terms. This distinguishes it from hyperbolas, which have a subtraction, and parabolas, which have only one squared term.
Recommended video:
5:30Foci and Vertices of an Ellipse
Related Practice
Textbook Question
27–32. Polar curves Graph the following equations.
r = 3 sin 4θ
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Textbook Question
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
c. The polar coordinates (3, -3π/4) and (-3, π/4) describe the same point in the plane.
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Textbook Question
22–23. Arc length Find the length of the following curves.
x = cos 2t, y = 2t - sin 2t; 0 ≤ t ≤ π/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.
a. A set of parametric equations for a given curve is always unique.
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Textbook Question
7–8. Parametric curves and tangent lines
a. Eliminate the parameter to obtain an equation in x and y.
x = 8cos t + 1, y = 8sin t + 2, for 0 ≤ t ≤ 2π; t = π/3
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