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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 12.4.22
Chapter 12, Problem 12.4.22

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.


25y² - 4x² = 100

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Rewrite the given equation \(25y^{2} - 4x^{2} = 100\) in standard form by dividing both sides by 100 to isolate the terms: \(\frac{25y^{2}}{100} - \frac{4x^{2}}{100} = 1\).
Simplify the fractions to get \(\frac{y^{2}}{4} - \frac{x^{2}}{25} = 1\). This is the standard form of a hyperbola centered at the origin with the \(y^{2}\) term positive, indicating it opens vertically.
Identify the values of \(a^{2}\) and \(b^{2}\) from the equation: \(a^{2} = 4\) and \(b^{2} = 25\). Here, \(a\) corresponds to the distance from the center to each vertex along the \(y\)-axis, and \(b\) relates to the shape of the hyperbola.
Find the vertices of the hyperbola, which lie along the \(y\)-axis at \((0, \pm a)\), so the vertices are at \((0, \pm 2)\).
Calculate the focal distance \(c\) using the relationship \(c^{2} = a^{2} + b^{2}\), then find the coordinates of the foci at \((0, \pm c)\). Finally, write the equations of the asymptotes, which for a vertical hyperbola centered at the origin are \(y = \pm \frac{a}{b} x\).

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Key Concepts

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

Identification of Conic Sections from General Equations

Conic sections are curves obtained by intersecting a plane with a double-napped cone. The general second-degree equation Ax² + By² + Cx + Dy + E = 0 can represent a parabola, ellipse, or hyperbola depending on the signs and values of A and B. If A and B have opposite signs, the conic is a hyperbola; if they are equal and nonzero, it is a circle; if both are positive but unequal, it is an ellipse; if one variable is squared and the other is linear, it is a parabola.
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Geometries from Conic Sections

Standard Forms and Key Features of Hyperbolas

A hyperbola's standard form is (x-h)²/a² - (y-k)²/b² = 1 or (y-k)²/a² - (x-h)²/b² = 1, where (h,k) is the center. Key features include vertices (points where the hyperbola intersects its principal axis), foci (points inside the branches related to the definition of the hyperbola), and asymptotes (lines the hyperbola approaches but never touches). These features help in sketching and understanding the hyperbola's shape.
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Circles in Standard Form Example 1

Graphing and Analyzing Conic Sections

Graphing conic sections involves rewriting the equation in standard form, identifying the center, vertices, foci, and axes lengths, and plotting these points. For hyperbolas, asymptotes are found using slopes ±b/a or ±a/b depending on orientation. Labeling these elements accurately provides a clear visual representation and aids in understanding the conic's geometric properties.
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Related Practice
Textbook Question

77–80. Slopes of tangent lines Find all points at which the following curves have the given slope.


x = 2 + √t, y = 2 - 4t; slope = -8

Textbook Question

57–64. Graphing polar curves Graph the following equations. Use a graphing utility to check your work and produce a final graph.


r = sin 3θ

Textbook Question

33–40. Areas of regions Make a sketch of the region and its bounding curves. Find the area of the region.


The region inside one leaf of r = cos 3θ

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Textbook Question

37–48. Polar-to-Cartesian coordinates Convert the following equations to Cartesian coordinates. Describe the resulting curve.


r = 3 csc θ

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Textbook Question

49–52. Cartesian-to-polar coordinates Convert the following equations to polar coordinates.


y = x²

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.


10x² - 7y² = 140

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