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New Perspectives on Conic Sections: A Classification Approach

Explore the relationship between directrix, focus, and eccentricity in conic sections. See how ellipse shape changes with eccentricity. Discover how every ellipse can be derived from a circle. Uncover the surprise link between eccentricity and tilt angle in conic sections.

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New Perspectives on Conic Sections: A Classification Approach

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  1. 6.7 A New Look at Conic Sections

  2. Plane parallel to an edge Plane perpendicular to the axis

  3. p. 247 Ex.

  4. Ex. 2

  5. READ TO STUDENTS One way to define a conic section is to specify a line in the plane, called the directrix, and a point in the plane off of the line, called the focus. The conic section is then the set of all points whose eccentricity is the ratio of “distance to the focus”: “distance to the directrix”. It is easy to see that as the eccentricity of an ellipse approaches 1, the ellipse becomes skinnier. The formula for the ellipse also shows that every ellipse can be obtained by taking a circle in a plane, lifting it up and out, tilting it, and projecting it back into the plane. Surprise: the eccentricity is equal to the sine of the angle of this tilt! A circle has a tilt of zero.

  6. Conic Sections

  7. 6.7 A New Look at Conic Sections Classifying 2nd-Degree Equations

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