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Magazine Chapter 8: Problem 5
What can be said about the symmetry of the graph of an ellipse with center at the origin and foci along the \(y\) -axis?
Short Answer
Expert verified
The ellipse is symmetric about the x-axis, y-axis, and origin.
Step by step solution
01
Understand the Ellipse Equation
An ellipse centered at the origin with foci along the \(y\)-axis has the standard form equation: \[ \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 \] where \(a > b\). Here, \(a\) is the semi-major axis length along the \(y\)-axis, and \(b\) is the semi-minor axis length along the \(x\)-axis.
02
Define Symmetry in the Context of an Ellipse
Symmetry in the context of a graph indicates that the graph looks the same when viewed from different perspectives or through various transformations such as reflection across axes or rotation around a point.
03
Check for Symmetry about the Y-Axis
Reflecting the ellipse equation \( \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 \) across the \(y\)-axis involves replacing \(x\) with \(-x\). After substitution, we get the same equation \( \frac{(-x)^2}{b^2} + \frac{y^2}{a^2} = 1 \), which is identical to the original equation. Thus, the ellipse is symmetric with respect to the \(y\)-axis.
04
Check for Symmetry about the X-Axis
Reflecting the ellipse equation across the \(x\)-axis involves replacing \(y\) with \(-y\). After substitution, the equation \( \frac{x^2}{b^2} + \frac{(-y)^2}{a^2} = 1 \) is obtained, which is exactly the same as the original equation. Therefore, the ellipse is symmetric with respect to the \(x\)-axis as well.
05
Check for Symmetry about the Origin
To check symmetry about the origin, replace \(x\) with \(-x\) and \(y\) with \(-y\), resulting in the equation \( \frac{(-x)^2}{b^2} + \frac{(-y)^2}{a^2} = 1 \). This is the same as the original equation, confirming that the ellipse is symmetric with respect to the origin.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Ellipse Equation
An ellipse is a type of conic section that often suggests a stretched circle. When centered at the origin, its equation takes on a specific form written as: \[ \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 \]Here, \(a\) and \(b\) are crucial as they represent the lengths of the semi-major and semi-minor axes of the ellipse. Notably, \(a\) must be greater than \(b\) if the foci are along the \(y\)-axis, indicating that the ellipse is taller than it is wide.
- \(x^2\) and \(y^2\) terms signify that the curve is centered at the origin.
- The coefficients \(b^2\) and \(a^2\) determine the spread of the ellipse along the x-axis and the y-axis respectively.
Symmetry in Graphs
Symmetry in graphs reveals how shapes or objects can look consistent from different viewpoints or after certain geometric transformations. In essence, this means:
- A graph is symmetric if it mirrors across a specific line, like an axis.
- It can also be symmetric if it rotates upon a point and looks unchanged.
Reflection Across Axes
Reflecting a function or equation across an axis is a powerful tool to determine symmetry. Here's how it applies to ellipses:
- Reflection across the \(y\)-axis: Change \(x\) to \(-x\). If the resulting equation is the same, it confirms \(y\)-axis symmetry.
- Reflection across the \(x\)-axis: Change \(y\) to \(-y\). If the equation remains unchanged, this indicates symmetry about the \(x\)-axis.
Origin Symmetry
Origin symmetry implies that an entire graph remains unchanged even after being flipped in both horizontal and vertical directions. Specifically, this involves replacing \(x\) with \(-x\) and \(y\) with \(-y\). For an ellipse equation:\[\frac{(-x)^2}{b^2} + \frac{(-y)^2}{a^2} = 1\]Since this version is identical to the original equation, the ellipse must be origin symmetric.
- Origin symmetry reveals that all points are equidistant along axes relative to the origin.
- This type of symmetry contributes to the holistic, balanced appearance of an ellipse.
