Eccentricity Formula
The eccentricity formula is a mathematical expression that describes the shape of an ellipse or a hyperbola
The eccentricity formula is a mathematical expression that describes the shape of an ellipse or a hyperbola. The eccentricity, denoted by the letter “e,” determines how elongated or compressed the shape is.
For an ellipse, the eccentricity formula is:
e = sqrt(1 – (b^2 / a^2))
In this formula, “a” represents the length of the semi-major axis, and “b” represents the length of the semi-minor axis. The semi-major axis is the longest distance from the center of the ellipse to any point on its boundary, while the semi-minor axis is the shortest distance.
The eccentricity of an ellipse can range from 0 to 1. When e = 0, the ellipse is a circle. As e approaches 1, the ellipse becomes more elongated or “stretched out.”
For a hyperbola, the eccentricity formula is slightly different:
e = sqrt(1 + (b^2 / a^2))
Similar to the ellipse formula, “a” represents the distance from the center to a vertex, and “b” represents the distance from the center to the edge of the hyperbola along the transverse axis.
The eccentricity of a hyperbola is always greater than 1. As e increases, the hyperbola becomes more elongated or “flattened.”
The eccentricity is a fundamental parameter that helps to define the shape of these conic sections. Through its calculation, we can determine whether an ellipse is more circular or elongated and whether a hyperbola is more symmetric or flattened.
Remember to plug in the correct values for “a” and “b” in the formulas to find the eccentricity of a specific ellipse or hyperbola.
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