The latus rectum is a line segment that passes through the focus of a conic section and is perpendicular to the major axis. It is an important geometric property that helps characterize the shape and size of ellipses, hyperbolas, and parabolas.
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The length of the latus rectum is equal to the square of the semi-minor axis divided by the semi-major axis for an ellipse, and the square of the semi-major axis divided by the semi-minor axis for a hyperbola.
For a parabola, the latus rectum is equal to twice the distance between the focus and the directrix.
The latus rectum provides information about the eccentricity of a conic section, with a larger latus rectum corresponding to a less eccentric shape.
In polar coordinates, the latus rectum represents the distance from the origin to the point where the conic section intersects a line perpendicular to the major axis and passing through the focus.
Understanding the latus rectum is important for visualizing the shape and size of conic sections, as well as for deriving equations and solving problems involving these geometric shapes.
Review Questions
Explain the relationship between the latus rectum and the major and minor axes of an ellipse.
For an ellipse, the length of the latus rectum is equal to the square of the semi-minor axis divided by the semi-major axis. This means that the latus rectum provides information about the eccentricity of the ellipse, with a larger latus rectum corresponding to a less eccentric (more circular) shape. The latus rectum is perpendicular to the major axis and passes through the focal points of the ellipse.
Describe how the latus rectum is used to characterize the shape of a hyperbola.
In the case of a hyperbola, the latus rectum is equal to the square of the semi-major axis divided by the semi-minor axis. This relationship is the inverse of the ellipse, reflecting the more elongated shape of the hyperbola. The latus rectum is still perpendicular to the major axis and passes through the focal points, providing information about the eccentricity and overall geometry of the hyperbolic curve.
Analyze the role of the latus rectum in the equation of a conic section in polar coordinates.
$$r = \frac{p}{1 + e\cos\theta}$$ In this polar equation for a conic section, the parameter $p$ represents the latus rectum. The latus rectum determines the distance from the origin to the point where the conic section intersects a line perpendicular to the major axis and passing through the focus. This relationship between the latus rectum and the polar equation highlights the importance of understanding this geometric property when working with conic sections in polar coordinates.
A measure of how elongated or flattened an ellipse or hyperbola is, defined as the ratio of the distance between the foci to the length of the major axis.
Focal Point: The point(s) around which an ellipse or hyperbola is constructed, located on the major axis and equidistant from the vertices.