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Ch. 7 - Conic Sections
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 7 - Conic SectionsProblem 1
Chapter 8, Problem 1

Graph the ellipse and locate the foci. x236+y225=1\(\frac{x^2}{36}\) + \(\frac{y^2}{25}\) = 1

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1
Identify the standard form of the ellipse equation: \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). Here, \(a^2 = 36\) and \(b^2 = 25\).
Determine the values of \(a\) and \(b\) by taking the square roots: \(a = \sqrt{36} = 6\) and \(b = \sqrt{25} = 5\).
Since \(a > b\), the major axis is along the x-axis. The ellipse is centered at the origin \((0,0)\) with vertices at \((\pm a, 0)\), which are \((\pm 6, 0)\).
Calculate the focal distance \(c\) using the relationship \(c^2 = a^2 - b^2\). Substitute the values to find \(c^2 = 36 - 25\).
Locate the foci at \((\pm c, 0)\) along the x-axis. These points are inside the ellipse between the center and the vertices.

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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 in standard form is written as (x^2/a^2) + (y^2/b^2) = 1, where a and b are the lengths of the semi-major and semi-minor axes. Identifying a and b helps determine the shape and orientation of the ellipse on the coordinate plane.
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Graph Ellipses at Origin

Graphing an Ellipse

To graph an ellipse, plot the center at the origin, then mark points a units along the major axis and b units along the minor axis. Connecting these points smoothly forms the ellipse, showing its size and orientation.
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Graph Ellipses NOT at Origin

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 c, found using c^2 = a^2 - b^2. Knowing c allows you to place the foci accurately on the graph.
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Foci and Vertices of an Ellipse
Related Practice
Textbook Question

Find the focus and directrix of each parabola with the given equation. Then match each equation to one of the graphs that are shown and labeled (a)–(d).

a. b. c. d.

y2 = 4x

Textbook Question

Graph each ellipse and locate the foci. x2/16+y2/4 = 1

Textbook Question

Find the vertices and locate the foci of each hyperbola with the given equation. Then match each equation to one of the graphs that are shown and labeled (a)–(d).

a. b. c. d.

x2/4−y2/1=1