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Understanding Ellipses: Equations, Properties, and Center

Learn about ellipses and their equations, properties such as vertices, covertices, foci, and eccentricity. Understand shifts to find standard ellipse equations.

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Understanding Ellipses: Equations, Properties, and Center

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  1. Section 11.3 The Ellipse

  2. THE ELLIPSE An ellipse is the set of all points in the plane, the sum of whose distances from two fixed points (foci) is a positive constant.

  3. if the major axis is horizontal. if the major axis is vertical. AN ELLIPSE WITH CENTER AT THE ORIGIN Using the distance formula and algebra, we can show that the equation for an ellipse whose center is the origin is:

  4. if the major axis is horizontal. if the major axis is vertical. ELLIPSES WITH CENTERAT (h, k) Using shifts (Section 2.5), we can find the standard form of an equation of an ellipse with center at (h, k).

  5. ELLIPSE FACTS • center: (h, k) • vertices: a units from center in both directions • covertices: b units from center in both directions • foci: c units from center in both directions where c2 = a2 – b2. • eccentricity: e = c/a. The eccentricity measures how far an ellipse is from a circle.

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