PreCalculus
Activity:
Properties of an Ellipse
Posted on Feb 25, 2010
Topic: Conics
Students discover properties of an ellipse, such as the set of all points such that the sum of the distances from these points to two fixed points is constant.
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Activity Key Steps:
Students investigate the definition of an ellipse. They graph an ellipse, trace the graph, and record the coordinates of at least ten points.
They will see that the sum of lengths PF1 and PF2 remain constant, while the individual lengths continuously change. From their observations, students discuss their findings and form their own definition of an ellipse.
Then, students examine the reflective properties of an ellipse. The screenshot to the right demonstrates that a tangent to the ellipse at point P was used to construct the rays that enter and leave point P.
Students will see that the outgoing ray is a reflection of the incoming ray. They observe this effect and recognize that any ray leaving one focus is reflected off the ellipse and directed to pass through the second focus.
At the end of the activity, students can identify an ellipse by definition and identify it when various planes intersect a cone. They will also be able to derive the equation of an ellipse and write its equation with a center (0, 0) given its vertices and co-vertices.