Textbook Question
A polar conic section Consider the equation r² = sec2θ
a. Convert the equation to Cartesian coordinates and identify the curve.
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16. Parametric Equations & Polar Coordinates
Conic Sections
2:48 minutesProblem 12.R.55c
Textbook Question
53–57. Conic sections
c. Find the eccentricity of the curve.
y² - 4x² = 16
Verified step by step guidance1
Rewrite the given equation \(y^{2} - 4x^{2} = 16\) in the standard form of a conic section by dividing both sides by 16, resulting in \(\frac{y^{2}}{16} - \frac{4x^{2}}{16} = 1\), which simplifies to \(\frac{y^{2}}{16} - \frac{x^{2}}{4} = 1\).
Recognize that the equation is in the form \(\frac{y^{2}}{a^{2}} - \frac{x^{2}}{b^{2}} = 1\), which represents a hyperbola centered at the origin with the transverse axis along the y-axis.
Identify the values of \(a^{2}\) and \(b^{2}\) from the equation: \(a^{2} = 16\) and \(b^{2} = 4\).
Recall the formula for the eccentricity \(e\) of a hyperbola: \(e = \frac{c}{a}\), where \(c\) is the distance from the center to a focus, and \(c^{2} = a^{2} + b^{2}\).
Calculate \(c\) using \(c^{2} = a^{2} + b^{2}\), then find the eccentricity \(e = \frac{c}{a}\). This will give the eccentricity of the hyperbola.
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2mKey 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, parabolas, or hyperbolas. Each conic has a standard equation form; for example, hyperbolas often appear as x²/a² - y²/b² = 1 or y²/a² - x²/b² = 1. Recognizing the type of conic from its equation is essential for further analysis.
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Parabolas as Conic Sections
Eccentricity of a Conic Section
Eccentricity (e) measures how much a conic deviates from being circular. For ellipses, 0 < e < 1; for parabolas, e = 1; and for hyperbolas, e > 1. It is defined as the ratio of the distance from a focus to a point on the curve over the perpendicular distance from that point to the directrix, and it characterizes the shape of the conic.
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Rewriting and Identifying Parameters from the Given Equation
To find the eccentricity, the given equation y² - 4x² = 16 must be rewritten in standard form by dividing both sides by 16, yielding (y²/16) - (x²/4) = 1. This identifies a hyperbola with a² = 16 and b² = 4, allowing calculation of eccentricity using e = √(1 + b²/a²). Understanding this manipulation is key to solving the problem.
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