1ZGeomDefns

Geometrical Definitions of Conics (not in course)

Circle

You are familiar with the concept of a circle having a center and a constant radius. In slightly more formal language:

A circle is defined as the set of points, P, around a focus (foci), F, such that the distance FP is constant.

ie FP = r, where r is the radius of the circle.

Ellipse An ellipse is defined as the set of points, P, around a pair of focuses (focii), F1 and F2, such that the __sum__ of the two distances, F 1 P + F 2 P is constant.

ie F 1 P +F 2 P = 2a, where 2a is the length of the major axis of the ellipse.

Hyperbola A hyperbola is defined as the set of points, P, around a pair of focuses (focii), F1 and F2, such that the __difference__ of the two distances, F 1 P – F 2 P is constant.

ie |F 1 P – F 2 P| = 2a, where 2a is the distance between the two vertices (closest points of the hyperbola).

Note: F 1 P – F 2 P = 2a creates the right branch F 2 P – F 1 P = 2a creates the left branch

Parabola A parabola is defined as the set of points, P, around a focus (foci) F, and a straight line (called a directrix), such that the distance FP is always equal to the distance from P to D where D is the closest point on the directrix.

ie FP = DP

Note that PD is perpendicular to the directrix.

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