Textbook Question
63–66. Tracing hyperbolas and parabolas Graph the following equations. Then use arrows and labeled points to indicate how the curve is generated as θ increases from 0 to 2π.
r = 3/(1 - cos θ)
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16. Parametric Equations & Polar Coordinates
Conic Sections
2:41 minutesProblem 12.R.56c
Textbook Question
53–57. Conic sections
c. Find the eccentricity of the curve.
x²/4 + y²/25 = 1
Verified step by step guidance1
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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Video duration:
2mKey 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.
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Parameterizing Equations of Circles & Ellipses
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.
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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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