Algebra II
Activity:
Constructing an Ellipse
Posted on May 12, 2008
Topic: Analytic Geometry
In this activity, students will explore two different methods for constructing an ellipse. Students discover that the sum of the distances from a point on the ellipse to its foci is always constant. This fact is then used as the basis for an algebraic derivation of the general equation for an ellipse centered at the origin.
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Activity Key Steps:
In problem 1, students explore the envelope construction method to construct an ellipse.
Students will see that point F lies on a diameter of the circle, segment FP connects F to the circle, and a perpendicular line to FP is drawn. They drag point P and create a locus of the perpendicular line as P travels along the circle, generating the shape of an ellipse.
The ellipse is formed by an envelope of lines hence the method’s name.
Students see that the diameter of the circle is equal to the width of the ellipse along its longer axis called the major axis. F is a special type of fixed point that can be used to generate the ellipse, called a focus.
Problem 2 gives students the opportunity to explore a second method called the String and Pins Construction. They use the definition of an ellipse to construct it. At the end of this activity, students will be able to derive the equation of an ellipse in rectangular form and write an equation of an ellipse with the center at (0, 0) given its vertices and co-vertices. They will also know two methods of constructing an ellipse.