/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 29 Graph each ellipse. $$ \frac... [FREE SOLUTION] | 91影视

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Chapter 13: Problem 29

Graph each ellipse. $$ \frac{x^{2}}{9}+\frac{y^{2}}{25}=1 $$

Short Answer

Expert verified
Draw an ellipse centered at (0,0) with major axis 10 units along y, and minor axis 6 units along x.

Step by step solution

01

- Identify the Standard Form

The given equation of the ellipse is: \[\frac{x^{2}}{9}+\frac{y^{2}}{25}=1.\]This matches the standard form of an ellipse: \[\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1.\]
02

- Determine the Values of a and b

Comparing the given equation with the standard form, we can identify the values of \(a\) and \(b\): \(a^{2} = 9\), so \(a = 3\) \(b^{2} = 25\), so \(b = 5\).
03

- Identify the Lengths of Major and Minor Axes

The length of the major axis is \(2b = 2(5) = 10\) units since \(b\) corresponds to the larger denominator. The length of the minor axis is \(2a = 2(3) = 6\) units.
04

- Identify the Center and Vertices

The center of the ellipse is at the origin, \((0, 0)\). The vertices on the major axis are \((0, \,\pm b) = (0, \, \pm 5)\) and on the minor axis they are \((\pm a, \, 0) = (\pm 3, \, 0)\).
05

- Draw the Axes

First, draw the horizontal axis (minor axis) with points at \( (\pm 3, 0) \). Then draw the vertical axis (major axis) with points at \( (0, \pm 5) \).
06

- Sketch the Ellipse

Using the vertices, sketch a smooth, oval-like shape ensuring it's symmetric with respect to both axes.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Standard Form of Ellipse
Understanding the standard form of an ellipse is crucial. It is written as: \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]. Here, \{a\} and \{b\} are the lengths of the ellipse's semi-axes. The squared terms indicate symmetry about the center. When graphing an ellipse, ensuring the equation matches this form helps us identify other components like the axes lengths and vertices accurately.
Major Axis and Minor Axis
The major axis of an ellipse is the longest diameter, extending through its foci. The minor axis, the shortest diameter, is perpendicular to the major axis and runs through the center. In the given ellipse equation \[ \frac{x^2}{9} + \frac{y^2}{25} = 1 \], the larger denominator identifies the length along the major axis, making \{y\}-axis the major axis. This axis extends from \{-b\} to \{+b\}, while the minor axis runs from \{-a\} to \{+a\} along the \{x\}-axis.
Vertices of Ellipse
Vertices are critical points located at the ends of the major and minor axes. For the standard form \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \], the vertices along the major axis are at (0, \{ \pm b \}) and along the minor axis at (\{ \pm a \}, 0). In our example, the peaks of the major axis are (0, \pm 5) and the minor axis' peaks at (\

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