Ch. 7 - Conic Sections

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 7 - Conic Sections

Problem 37

Chapter 8, Problem 37

Graph each ellipse and give the location of its foci. (x − 2)²/9 + (y -1)² /4= 1

Verified step by step guidance

Identify the standard form of the ellipse equation: \(\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1\), where \((h, k)\) is the center of the ellipse.

From the given equation \(\frac{(x - 3)^2}{36} + \frac{(y - 4)^2}{25} = 1\), determine the center as \((3, 4)\).

Compare the denominators to find \(a^2\) and \(b^2\). Since 36 > 25, \(a^2 = 36\) and \(b^2 = 25\). This means the major axis is horizontal because \(a^2\) is under the \((x - h)^2\) term.

Calculate the lengths of the semi-major axis \(a = \sqrt{36}\) and the semi-minor axis \(b = \sqrt{25}\).

Find the distance \(c\) from the center to each focus using the formula \(c = \sqrt{a^2 - b^2}\). Then, locate the foci at \((h \pm c, k)\) because the major axis is horizontal.

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Key Concepts

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

Standard Form of an Ellipse

The standard form of an ellipse equation is (x-h)²/a² + (y-k)²/b² = 1, where (h, k) is the center. The values a² and b² represent the squares of the lengths of the semi-major and semi-minor axes, respectively. Understanding this form helps in identifying the ellipse's size, shape, and position on the coordinate plane.

Major and Minor Axes

The major axis is the longest diameter of the ellipse, while the minor axis is the shortest. The lengths of these axes are 2a and 2b, where a and b are the square roots of the denominators in the ellipse equation. Knowing which axis is major or minor depends on whether a² > b² or vice versa.

Foci of an Ellipse

The foci are two fixed points inside the ellipse such that the sum of distances from any point on the ellipse to the foci is constant. Their locations are found using c² = a² - b², where c is the distance from the center to each focus along the major axis. Identifying the foci is essential for graphing and understanding ellipse properties.

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Foci and Vertices of an Ellipse

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