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Chapter 12: Problem 21

Identify the graph of each equation as a parabola, circle, ellipse, or hyperbola, and then sketch the graph. $$ x^{2}+9 y^{2}=9 $$

Short Answer

Expert verified
The graph is an ellipse. The axes lengths are 6 (x-axis) and 2 (y-axis).

Step by step solution

01

Identify the General Form

Compare the given equation with standard forms of conic sections. The given equation is \[ x^{2} + 9 y^{2} = 9 \].
02

Rewrite the Equation

Rewrite the equation in standard form for an ellipse. Divide every term by 9: \[ \frac{x^2}{9} + \frac{9y^2}{9} = \frac{9}{9} \] which simplifies to \[ \frac{x^2}{9} + y^2 = 1 \].
03

Recognize the Conic Section

The equation \[ \frac{x^2}{9} + y^2 = 1 \] represents an ellipse because it fits the standard form \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \], where \( a = 3 \) and \( b = 1 \).
04

Sketch the Graph

Draw the ellipse. Identify the vertices and co-vertices: - The major axis is along the x-axis with length \( 2a = 6 \) (vertices at \( (3,0) \) and \( (-3,0) \)). - The minor axis is along the y-axis with length \( 2b = 2 \) (co-vertices at \( (0,1) \) and \( (0,-1) \)).

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Key Concepts

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

Conic Sections
Conic sections are curves generated by slicing a cone at different angles. These include parabolas, circles, ellipses, and hyperbolas.

Each type of conic section has a standard form equation that helps identify the conic type from the given equation. In our case, we're examining an ellipse.

Recognizing whether a conic section is a circle, ellipse, hyperbola, or parabola involves inspecting the coefficients and their relationship in the equation.
Ellipse Equation
The standard form of the ellipse equation is \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). Here, \( a^2 \) and \( b^2 \) are the denominators which define the lengths of the axes.

The equation \( x^2+9y^2=9 \) can be rewritten as \( \frac{x^2}{9} + y^2 = 1 \), which fits the standard form.

In this equation, \( a^2=9 \) and \( b^2=1 \). Thus, \( a=3 \) and \( b=1 \). This helps us understand the ellipse's dimensions.
Graphing Ellipses
Graphing ellipses involves understanding the major and minor axes. The equation tells us where to place the vertices and co-vertices.

For \( \frac{x^2}{9} + y^2 = 1 \), we identify:
  • Major axis: runs along the x-axis because \( a=3 \)
  • Minor axis: runs along the y-axis because \( b=1 \)

To draw:
  • Plot points at \( (3,0) \) and \( (-3,0) \)
  • Plot minor axis points at \( (0,1) \) and \( (0,-1) \)
  • Connect these points to form the ellipse shape

Always label the axes to help visualize the ellipse orientation.
Major and Minor Axes
In ellipses, the longer axis is the major axis, and the shorter one is the minor axis. They determine the ellipse's width and height.

The length of the major axis is \( 2a \), and for the given ellipse, \( a=3 \), so the major axis spans 6 units.

The minor axis length is \( 2b \). For our ellipse, \( b=1 \), hence it spans 2 units.

These axes intersect at the ellipse's center and help graph the ellipse accurately.

In summary:
  • Major axis: length 6 units
  • Minor axis: length 2 units
Understanding these axes is crucial for correctly plotting and interpreting ellipses.

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