Textbook Question
Finding Parametric Equations and Tangent Lines
Find parametric equations for the given curve.
9x² + 4y² = 36
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Ch. 11 - Parametric Equations and Polar Coordinates
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 11, Problem 11.PE.62
Graphing Conic Sections
Find the eccentricities of the ellipses and hyperbolas in Exercises 59–62. Sketch each conic section. Include the foci, vertices, and asymptotes (as appropriate) in your sketch.
5y² − 4x² = 20
Verified step by step guidance1
Rewrite the given equation in the standard form of a conic section. Start by dividing both sides of the equation by 20 to isolate the terms: \(\frac{5y^{2}}{20} - \frac{4x^{2}}{20} = 1\) which simplifies to \(\frac{y^{2}}{4} - \frac{x^{2}}{5} = 1\).
Identify the type of conic section from the standard form. Since the equation is of the form \(\frac{y^{2}}{a^{2}} - \frac{x^{2}}{b^{2}} = 1\), it represents a hyperbola that opens vertically (along the y-axis).
Recall the formula for the eccentricity \(e\) of a hyperbola: \(e = \frac{c}{a}\), where \(c^{2} = a^{2} + b^{2}\). Here, \(a^{2} = 4\) and \(b^{2} = 5\). Calculate \(c\) by finding \(c = \sqrt{a^{2} + b^{2}}\).
Determine the coordinates of the vertices and foci. For a hyperbola opening vertically, vertices are at \((0, \pm a)\) and foci at \((0, \pm c)\). Plot these points accordingly.
Find the equations of the asymptotes, which for a vertical hyperbola are given by \(y = \pm \frac{a}{b} x\). Use the values of \(a\) and \(b\) to write these equations and include them in your sketch.
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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 ellipses, hyperbolas, parabolas, or circles. Each conic has a standard equation form; for example, ellipses and hyperbolas have distinct forms involving squared terms with positive or negative signs. Recognizing and rewriting the given equation into a standard form is essential for identifying the conic type and its properties.
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Parabolas as Conic Sections
Eccentricity of Ellipses and Hyperbolas
Eccentricity (e) measures how much a conic deviates from being circular. For ellipses, e ranges between 0 and 1, calculated as the ratio of the distance between foci to the major axis length. For hyperbolas, e is greater than 1, reflecting their open shape. Calculating eccentricity involves using the relationship between the semi-major axis, semi-minor axis, and focal distance.
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6:15Introduction to Hyperbolas
Graphing Conic Sections: Foci, Vertices, and Asymptotes
Graphing conics requires plotting key features: vertices (points where the conic intersects its principal axis), foci (fixed points defining the conic), and asymptotes (lines that hyperbolas approach at infinity). For ellipses, asymptotes are not present, but for hyperbolas, asymptotes guide the shape. Identifying these elements helps in accurately sketching the conic section.
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Related Practice
Textbook Question
Graphing Conic Sections
Exercises 63-68 give equations for conic sections and tell how many units up or down and to the right or left each curve is to be shifted. Find an equation for the new conic section, and find the new foci, vertices, centers, and asymptotes, as appropriate. If the curve is a parabola, find the new directrix as well.
x²/169 + y²/144 = 1, right 5, up 12
Textbook Question
Lines
Sketch the lines in Exercises 45–48 and find Cartesian equations for them.
r cos (θ + π/3) = 2
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Textbook Question
Identifying Parametric Equations in the Plane
Exercises 1–6 give parametric equations and parameter intervals for the motion of a particle in the xy-plane. Identify the particle’s path by finding a Cartesian equation for it. Graph the Cartesian equation and indicate the direction of motion and the portion traced by the particle.
x = 4 cos t, y = 9 sin t, 0 ≤ t ≤ 2π
Textbook Question
Polar Coordinates
Exercises 19–22 give the eccentricities of conic sections with one focus at the origin of the polar coordinate plane, along with the directrix for that focus. Find a polar equation for each conic section.
e = 1/3, r sin θ = −6
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Textbook Question
Finding Parametric Equations and Tangent Lines
Find parametric equations for the given curve.
Line through (1,-2) with slope 3