Precalculus: Foci Definition of Ellipses and Hyperbolas

Objectives

• Students will be able to define an ellipse as the set of points whose distances to two fixed points (foci) have a constant sum.
• Students will be able to define a hyperbola as the set of points whose distances to two fixed points (foci) have a constant difference.
• Students will be able to describe the relationship between the location of the foci and the shapes of the corresponding ellipses and hyperbolas.
• Students will be able to determine the effect of the eccentricity of ellipses and hyperbolas on the shape of their curves.

Vocabulary

• conjugate axis
• eccentricity
• ellipse
• focus/foci
• hyperbola
• major axis
• minor axis
• semi-major axis
• semi-minor axis
• transverse axis
• vertex of a conic

About the Lesson

This lesson involves observing and describing the relationships between the foci of ellipses and hyperbolas and the shape of the corresponding curves.
As a result, students will:

• Define an ellipse as the set of points whose distances to two fixed points (foci) have a constant sum.
• Define a hyperbola as the set of points whose distances to two fixed points (foci) have a constant difference.
• Manipulate sliders to observe the relationship between the foci and sum/difference of the distances from the foci to a point on the curve.
• Observe the effect of the relationship between the foci and the shapes of ellipses or hyperbolas.