Question

What is the eccentricity of a conic section? How can you classify conic sections by eccentricity? How does eccentricity change the shape of ellipses and hyperbolas?

Conic sections
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asked 2020-11-09
What is the eccentricity of a conic section? How can you classify conic sections by eccentricity? How does eccentricity change the shape of ellipses and hyperbolas?

Answers (1)

2020-11-10
Step 1 Any conic section can be defined as the locus of points whose distances to a point (the focus) and a line (the directrix) are in a constant ratio. That ratio is called the eccentricity of the conic section , which is often denoted as e. Step 2 The eccentricity (e) of an ellipse which is not a circle is greater than zero but less than 1. If \(e = 1\) then it is Parabola. If \(e > 1\) then it is Hyperbola [The eccentricity of a hyperbola can be any real number greater than 1, with no upper bound. The eccentricity of a rectangular hyperbola is \(\sqrt20\). If \(e = 0\) then it is Circle. Step 3 If an ellipse is close to circular it has an eccentricity close to zero. If an ellipse has an eccentricity close to one it has a high degree of ovalness. The eccentricity of a conic section tells us how close it is to being in the shape of a circle. The farther away the eccentricity of a conic section is from 0, the less the shape looks like a circle. an ellipse looks like a compressed circle. When compared with the other two conic shapes, it most closely resembles a circle. Similar reasoning deduces that a parabola would be next closest, and a hyperbola the farthest from a circle in shape.
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