Calculating Eccentricity: Understanding the Formula & Its Applications in Geometry

Eccentricity Formula

The eccentricity formula is used in geometry to calculate the eccentricity of certain conic sections, particularly ellipses and hyperbolas

The eccentricity formula is used in geometry to calculate the eccentricity of certain conic sections, particularly ellipses and hyperbolas. The eccentricity of a conic section determines its shape.

For an ellipse, the eccentricity (ε) is calculated using the formula:

ε = √(1 – (b^2/a^2))

where a and b are the lengths of the semi-major and semi-minor axes, respectively. The semi-major axis is the longest distance from the center of the ellipse to the edge, while the semi-minor axis is the shortest distance.

For a hyperbola, the eccentricity (ε) is calculated using the formula:

ε = √(1 + (b^2/a^2))

Again, a and b are the lengths of the semi-major and semi-minor axes.

In both formulas, if ε is greater than 1, the conic section is a hyperbola. If ε is equal to 1, the conic section is a parabola. If ε is less than 1, the conic section is an ellipse.

Let’s take an example to illustrate the use of the eccentricity formula. Suppose we have an ellipse with a semi-major axis of 6 units and a semi-minor axis of 4 units. To find the eccentricity, we can use the formula:

ε = √(1 – (4^2/6^2))

ε = √(1 – 16/36)

ε = √(1 – 4/9)

ε = √(5/9)

ε ≈ 0.745

Since the calculated eccentricity, 0.745, is less than 1, we can conclude that the conic section is an ellipse.

Remember that the eccentricity formula is specific to ellipses and hyperbolas. It helps in determining the shape of these conic sections and provides a measure of how elongated or squished they are.

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