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Chapter 13: Problem 48

Describe how to graph \(\frac{x^{2}}{25}+\frac{y^{2}}{16}=1\)

Short Answer

Expert verified
The graph of \(\frac{x^{2}}{25}+\frac{y^{2}}{16}=1\) is an ellipse centered at (0,0), with a major axis length of 10 units (it extends 5 units to the left and right of the center) and a minor axis length of 8 units (extending 4 units above and below the center).

Step by step solution

01

Identify the type of conic section

Identify the given equation as an ellipse since it is in the form \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\).
02

Find the center of the ellipse

The standard equation of an ellipse in the format given is centered at the origin, (0,0), so in this case the center of the ellipse is at (0,0).
03

Identify the values of a and b

This is determined by comparing the given equation to the standard form, thus we identify \(a^{2} = 25\) and \(b^{2} = 16\). Meaning, \(a = 5\) and \(b = 4\).
04

Plot the ellipse

First, plot the center of the ellipse at the point (0,0). Then, plot points at a distance \(a = 5\) units to the left and right of the center along the x-axis, and at a distance \(b = 4\) units above and below the center along the y-axis. Finally, sketch the ellipse by drawing a smooth curve that connects the points, making sure it extends along the longer (major) axis.

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