Ch.12 - Parametric and Polar Curves

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch.12 - Parametric and Polar Curves

Problem 12.R.56c

Chapter 12, Problem 12.R.56c

53–57. Conic sections
c. Find the eccentricity of the curve.
x²/4 + y²/25 = 1

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Identify the type of conic section given by the equation \(\frac{x^{2}}{4} + \frac{y^{2}}{25} = 1\). Since both \(x^{2}\) and \(y^{2}\) terms are positive and the equation equals 1, this is an ellipse.

Recall the standard form of an ellipse centered at the origin: \(\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1\), where \(a\) and \(b\) are the semi-major and semi-minor axes. Determine which denominator is larger to identify \(a^{2}\) and \(b^{2}\).

Assign \(a^{2} = 25\) and \(b^{2} = 4\) because 25 is greater than 4, so \(a = 5\) and \(b = 2\). The major axis is along the \(y\)-axis since \(a^{2}\) is under \(y^{2}\).

Use the formula for the eccentricity \(e\) of an ellipse: \(e = \frac{c}{a}\), where \(c\) is the distance from the center to a focus. Calculate \(c\) using \(c^{2} = a^{2} - b^{2}\).

Substitute the values of \(a^{2}\) and \(b^{2}\) into \(c^{2} = a^{2} - b^{2}\) to find \(c\), then compute \(e = \frac{c}{a}\). This will give the eccentricity of the ellipse.

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Key Concepts

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

Equation of an Ellipse

An ellipse is defined by the equation (x²/a²) + (y²/b²) = 1, where a and b are the lengths of the semi-major and semi-minor axes. Identifying which axis is major or minor depends on the relative sizes of a and b. This form helps in understanding the shape and orientation of the ellipse.

Eccentricity of an Ellipse

Eccentricity (e) measures how much an ellipse deviates from being a circle, calculated as e = √(1 - (b²/a²)) when a > b. It ranges from 0 (circle) to 1 (parabola). Knowing eccentricity helps describe the ellipse's shape and its geometric properties.

Identifying Major and Minor Axes

In the ellipse equation, the larger denominator corresponds to the square of the semi-major axis (a²), and the smaller to the semi-minor axis (b²). Correctly identifying these axes is essential for calculating eccentricity and understanding the ellipse's geometry.

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Related Practice

Textbook Question

45–60. Areas of regions Find the area of the following regions.

The region inside the lemniscate r² = 6 sin 2θ

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65–68. Eccentricity-directrix approach Find an equation of the following curves, assuming the center is at the origin. Graph the curve, labeling vertices, foci, asymptotes (if they exist), and directrices.

A hyperbola with vertices (0, ±2) and directrices y = ±1

Textbook Question

3–6. Eliminating the parameter Eliminate the parameter to find a description of the following curves in terms of x and y. Give a geometric description and the positive orientation of the curve.

x = sin t - 3, y = cos t + 6; 0 ≤ t ≤ π

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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.

Textbook Question

53–57. Conic sections

a. Determine whether the following equations describe a parabola, an ellipse, or a hyperbola.