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Ch. 7 - Conic Sections
Blitzer - College Algebra 8th Edition
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Tutti i libri di testoBlitzer 8th EditionCh. 7 - Conic SectionsProblema 39
Capitolo 8, Problema 39

Graph each ellipse and give the location of its foci. (x +3)²+ 4(y -2)² = 16

Guida verificata passo dopo passo
1
Identify the standard form of the ellipse equation. The given equation is \( (x + 1)^2 + 4(y + 4)^2 = 100 \). To write it in the standard form of an ellipse, divide both sides by 100 to get \( \frac{(x + 1)^2}{100} + \frac{4(y + 4)^2}{100} = 1 \).
Simplify the second term by dividing 4 by 100: \( \frac{(x + 1)^2}{100} + \frac{(y + 4)^2}{25} = 1 \). Now the equation is in the form \( \frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1 \), where the center is at \((-1, -4)\).
Determine the values of \(a^2\) and \(b^2\). Here, \(a^2 = 100\) and \(b^2 = 25\). Since \(a^2 > b^2\), the major axis is horizontal, and the ellipse stretches more along the x-axis.
Find the lengths of the semi-major axis \(a\) and semi-minor axis \(b\) by taking the square roots: \(a = \sqrt{100} = 10\) and \(b = \sqrt{25} = 5\).
Calculate the focal distance \(c\) using the relationship \(c^2 = a^2 - b^2\). Then, find the coordinates of the foci by moving \(c\) units left and right from the center along the major axis (x-axis). The foci are at \((-1 - c, -4)\) and \((-1 + c, -4)\).

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Standard Form of an Ellipse

The standard form of an ellipse equation is (x-h)²/a² + (y-k)²/b² = 1 or (x-h)²/b² + (y-k)²/a² = 1, where (h, k) is the center. It shows how the ellipse is stretched along the x- and y-axes. Recognizing this form helps identify the ellipse's center and the lengths of its axes.
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Graph Ellipses at Origin

Identifying the Center and Axes Lengths

From the equation, the center is at (-1, -4) by reversing the signs inside the parentheses. The denominators under the squared terms determine the lengths of the semi-major and semi-minor axes. Dividing both sides by 100 puts the equation in standard form, revealing these lengths.
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Identifying Intervals of Unknown Behavior

Finding the Foci of an Ellipse

The foci lie along the major axis, located at a distance c from the center, where c² = a² - b². Here, a is the semi-major axis length and b the semi-minor axis length. Calculating c allows you to find the exact coordinates of the foci, essential for graphing the ellipse accurately.
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Foci and Vertices of an Ellipse
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