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Chapter 12: Problem 30

Use a graphing device to graph the ellipse. $$ x^{2}+\frac{y^{2}}{12}=1 $$

Short Answer

Expert verified
The ellipse is centered at (0,0) with the semi-major axis along the y-axis, extending to \( 2\sqrt{3} \), and the semi-minor axis along the x-axis, extending to 1.

Step by step solution

01

Understand the Structure of the Ellipse Equation

The given equation of the ellipse is \( x^2 + \frac{y^2}{12} = 1 \). This can be rewritten in the standard form of an ellipse, \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), where \( a \) and \( b \) are the semi-major and semi-minor axes, respectively.
02

Identify the Semi-Axes

Compare \( x^2 + \frac{y^2}{12} = 1 \) with the standard form. Here, \( a^2 = 1 \) and \( b^2 = 12 \), hence \( a = 1 \) and \( b = \sqrt{12} = 2\sqrt{3} \). The ellipse is centered at the origin (0,0) with semi-major axis \( b = 2\sqrt{3} \) along the y-axis and semi-minor axis \( a = 1 \) along the x-axis.
03

Set Up the Graphing Device

Using a graphing device, set the equation \( x^2 + \frac{y^2}{12} = 1 \). Ensure that the scale for the x-axis and y-axis is appropriately set to visually represent the semi-major and semi-minor axes.
04

Graph the Ellipse

Plot the ellipse on the graph. The ellipse should stretch vertically along the y-axis up to \( 2\sqrt{3} \) and horizontally along the x-axis up to \( 1 \).
05

Verify the Plot

After plotting, confirm that the ellipse extends to \( x = 1 \) and \( y = 2\sqrt{3} \) at their respective maximum extents, and that the center is at the origin (0,0).

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Key Concepts

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

Understanding the Semi-Major Axis
In an ellipse, the semi-major axis is the longer one of the two axes. It is an important feature as it defines how stretched or elongated the ellipse is.
The semi-major axis sits along the direction where the ellipse is widest. For the equation \( x^2 + \frac{y^2}{12} = 1 \), the semi-major axis is determined by comparing it to the standard form of an ellipse equation \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \).
Here, \( b^2 = 12 \), so \( b = \sqrt{12} = 2\sqrt{3} \), indicating that the longer axis is along the y-axis.
Exploring the Semi-Minor Axis
The semi-minor axis is the shorter axis of the ellipse, indicating how narrow the ellipse is. In our given equation, this axis is aligned with the x-axis.
Comparing the equation to the standard form of an ellipse, we find that \( a^2 = 1 \), which means \( a = 1 \).
Thus, the semi-minor axis measures \( 1 \) unit on both sides of the center along the x-axis, shaping the ellipse's narrow dimension.
The Significance of the Standard Form of an Ellipse
The standard form of an ellipse equation is written as \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). This form is crucial as it allows us to visually understand and graph the ellipse, identifying both axes quickly.
In this setup:
  • \( a \) and \( b \) are lengths of semi-minor and semi-major axes.
  • The numbers under the \( x^2 \) and \( y^2 \) terms denote the squared lengths of the semi-axes.
  • Ensures that the ellipse is centered at the origin \((0,0)\) unless otherwise specified.
The form simplifies the process of plotting by indicating which axis is major based on the larger denominator.
Using a Graphing Device Effectively
Graphing devices help visualize mathematical concepts like ellipses in a clear, interactive manner. To plot an ellipse such as \( x^2 + \frac{y^2}{12} = 1 \) with a graphing device, follow these steps:
  • Input the equation into the graphing tool as given.
  • Adjust the axis scales using the semi-major and semi-minor axes to ensure accurate representation. The x-axis should extend to \( 1 \), while the y-axis reaches \( 2\sqrt{3} \).
  • Verify the plot by ensuring the ellipse is centered at the origin and aligns with identified parameters.
Utilizing these tools can enhance comprehension by providing a visual representation of abstract mathematical ideas.

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Most popular questions from this chapter

Find the vertices, foci, and asymptotes of the hyperbola, and sketch its graph. $$ 4 y^{2}-x^{2}=1 $$

Graph the conics \(r=e /(1-e \cos \theta)\) with \(e=0.4,0.6,0.8\) and 1.0 on a common screen. How does the value of \(e\) affect the shape of the curve?

(a) Show that the asymptotes of the hyperbola \(x^{2}-y^{2}=5\) are perpendicular to each other. (b) Find an equation for the hyperbola with foci \(( \pm c, 0)\) and with asymptotes perpendicular to each other.

Determine what the value of \(F\) must be if the graph of the equation $$4 x^{2}+y^{2}+4(x-2 y)+F=0$$ is (a) an ellipse, (b) a single point, or (c) the empty set.

\(23-34\) Complete the square to determine whether the equation represents an ellipse, a parabola, a hyperbola, or a degenerate conic. If the graph is an ellipse, find the center, foci, vertices, and lengths of the major and minor axes. If it is a parabola, find the vertex, focus, and directrix. If is a hyperbola, find the center, foci, vertices, and asymptotes. Then sketch the graph of the equation. If the equation has no graph, explain why. $$ 3 x^{2}+4 y^{2}-6 x-24 y+39=0 $$
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