Conic sections

Connect defining distance conditions with foci, equations, and graphs.

Meaning

Parabolas, ellipses, and hyperbolas have focal distance definitions. A parabola consists of points equidistant from a focus and a line called the directrix, with the focus not on the directrix. An ellipse has a constant sum of distances to two foci. A hyperbola has a constant absolute difference of distances to two foci.

Example

Points equidistant from focus and directrix form is distance 2 from both.For (x,y), the squared distance to the focus is x²+(y−1)²; the squared distance to the directrix is (y+1)². Equating them and simplifying gives x²=4y, hence y=x²/4.An ellipse's constant sum must exceed the distance between its foci. A hyperbola's constant absolute difference must be positive and smaller than that distance; boundary cases are not ordinary ellipses or hyperbolas.

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Scope and references

Topics

Parabolas, ellipses, hyperbolas, foci and basic properties

Learning goal

Connect defining distance conditions with foci, equations, and graphs.

Helpful background

Explore coordinates, distances, and equation-defined shapes.

Conic sections

These ordinary choice questions provide limited practice in this concept. They do not establish independent solution, full-concept performance, or mastery.

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Conic sections

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Conic sections