Ch. 7 - Conic Sections

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

Blitzer 8th Edition

Ch. 7 - Conic Sections

Problem 13

Chapter 8, Problem 13

Graph each ellipse and locate the foci. 25x²+4y² = 100

Verified step by step guidance

Rewrite the given equation of the ellipse in standard form by dividing both sides of the equation by 100: \(\frac{25x^{2}}{100} + \frac{4y^{2}}{100} = \frac{100}{100}\).

Simplify the fractions to get the equation in the form \(\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1\).

Identify the values of \(a^{2}\) and \(b^{2}\) from the simplified equation. Determine which is larger to know the orientation of the ellipse (horizontal or vertical).

Calculate the value of \(c\) using the relationship \(c^{2} = |a^{2} - b^{2}|\), where \(c\) is the distance from the center to each focus.

Locate the foci on the coordinate plane by placing them \(c\) units along the major axis from the center at the origin, and sketch the ellipse using the intercepts from \(a\) and \(b\).

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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

An ellipse equation can be written in the standard form (x²/a²) + (y²/b²) = 1, where a and b are the lengths of the semi-major and semi-minor axes. Converting the given equation into this form helps identify these axes and facilitates graphing the ellipse.

Identifying the Major and Minor Axes

The major axis is the longer axis of the ellipse, and the minor axis is the shorter one. By comparing a² and b² from the standard form, you determine which axis is major or minor, which is essential for correctly sketching the ellipse and locating its foci.

Locating the Foci of an Ellipse

The foci are two fixed points inside the ellipse located along the major axis. Their distance from the center is given by c, where c² = |a² - b²|. Knowing a, b, and c allows you to plot the foci accurately on the graph.

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Foci and Vertices of an Ellipse

Related Practice

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