Ch. 7 - Conic Sections

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

Blitzer 8th Edition

Ch. 7 - Conic Sections

Problem 15

Chapter 8, Problem 15

Graph each ellipse and locate the foci.4x²+16y² = 64

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Rewrite the given ellipse equation \(4x^2 + 16y^2 = 64\) in standard form by dividing every term by 64 to isolate 1 on the right side: \(\frac{4x^2}{64} + \frac{16y^2}{64} = 1\).

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

Determine which denominator is larger to identify the major axis. If \(a^2 > b^2\), the major axis is along the x-axis; if \(b^2 > a^2\), it is along the y-axis.

Calculate the coordinates of the vertices using \(\pm a\) or \(\pm b\) depending on the major axis. These points lie on the ellipse along the major axis.

Find the foci by calculating \(c = \sqrt{|a^2 - b^2|}\). The foci are located at \((\pm c, 0)\) if the major axis is horizontal, or \((0, \pm c)\) if vertical.

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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 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 understand the ellipse's shape and orientation.

Graphing an Ellipse

Graphing an ellipse involves plotting its center, major and minor axes based on a and b values, and sketching the curve accordingly. Knowing the lengths and orientation of these axes allows accurate representation of the ellipse on the coordinate plane.

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 positions are found using c² = |a² - b²|, where c is the distance from the center to each focus along the major axis.

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Foci and Vertices of an Ellipse

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