The minor axis of an ellipse is the shorter of the two principal axes that define the shape of the ellipse. It is the axis that runs perpendicular to the major axis, passing through the center of the ellipse.
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The minor axis of an ellipse is always perpendicular to the major axis and passes through the center of the ellipse.
The length of the minor axis, along with the length of the major axis, determines the eccentricity of the ellipse.
As the eccentricity of an ellipse increases, the minor axis becomes shorter relative to the major axis.
The minor axis is one of the key parameters used to describe the shape and size of an ellipse.
In the context of the Rotation of Axes, the minor axis is used to define the new coordinate system after the ellipse has been rotated.
Review Questions
Explain the relationship between the minor axis and the major axis of an ellipse.
The minor axis and major axis of an ellipse are the two principal axes that define the shape of the ellipse. The minor axis is the shorter of the two axes and runs perpendicular to the major axis, passing through the center of the ellipse. The lengths of the major and minor axes, along with the focal points, determine the eccentricity of the ellipse, which is a measure of how much the ellipse deviates from being a perfect circle.
Describe how the minor axis is used in the context of the Rotation of Axes topic.
In the Rotation of Axes topic, the minor axis is used to define the new coordinate system after an ellipse has been rotated. The minor axis, along with the major axis, is used to establish the orientation of the new coordinate system relative to the original coordinate system. This allows for the transformation of the equation of the ellipse from the original coordinate system to the new, rotated coordinate system, which can be useful in various applications involving ellipses.
Analyze how changes in the length of the minor axis affect the overall shape and eccentricity of an ellipse.
$$\text{As the length of the minor axis decreases relative to the major axis, the eccentricity of the ellipse increases.}$$ This means that the ellipse becomes more elongated and deviates further from being a perfect circle. Conversely, as the minor axis length increases, the eccentricity decreases, and the ellipse becomes more circular in shape. The minor axis, along with the major axis, is a key parameter in determining the overall shape and properties of an ellipse.
The major axis of an ellipse is the longer of the two principal axes that define the shape of the ellipse. It is the axis that runs through the center of the ellipse, with the longest distance between the two points where the ellipse intersects this axis.
Eccentricity is a measure of how much an ellipse deviates from being a perfect circle. It is a value between 0 and 1, with 0 representing a circle and values closer to 1 representing more elongated ellipses.
Focal Points: The focal points of an ellipse are two points within the ellipse that define its shape. The distance between these two points is called the focal length, and it is used to determine the eccentricity of the ellipse.