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Constructing an Ellipse

On this web page you can explore the geometric construction of an ellipse. In Lesson 9.2 of Discovering Advanced Algebra: An Investigative Approach, you learned the locus definition of an ellipse.

Definition of an Ellipse

An ellipse is a locus of points P in a plane, the sum of whose distances, d1 and d2, from two fixed points, F1 and F2, is always a constant, d. That is, d1 + d2 = d, or F1P + F2P = d. The two fixed points, F1 and F2, are called foci.

The sketches on this page use this definition to construct an ellipse. The construction is the same as the one explained in the Exploration Constructing the Conic Sections in Chapter 8.

Sketch

This sketch shows two circles centered at F1 and F2 and the points where they intersect. The radius of circle F1 is congruent to segment AC, and the radius of circle F2 is congruent to segment CB. Drag point C to trace the intersections of the circles as the radii change. Drag points F1 and F2 to reposition the foci, and drag point B to change the sum of the lengths of the radii. Click X to erase the traces.

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Investigate

  1. Slowly drag point C back and forth along segment AB. What shape is created by the trace of the intersections of the circles?
  2. What is true about the sum of the distance from an intersection point to F1 and the distance from the same intersection point to F2? In what ways is this sum represented by the sketch?
  3. Why is the trace of the intersections an ellipse?
  4. Try moving F1, F2, and/or B, erase the traces, then slowly drag C back and forth. Do you still get an ellipse? In what ways has it changed?

Sketch

This sketch shows the locus of the intersections from the previous sketch. Drag points F1 and F2 to reposition the foci, and drag point B to change the sum of the lengths of the radii. (Note: For technical reasons, there may be small gaps at the ends of the ellipse.)

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Investigate

  1. Experiment by dragging the foci farther apart and closer together. How does the ellipse change?
  2. How far apart can the foci be before you no longer have an ellipse?
  3. What happens when both foci are at the same point?
  4. Experiment by dragging point B. How does the ellipse change? Based on the locus definition of an ellipse, what is changing?
  5. How can you make the ellipse taller than it is wide?

Go to the parabola page or the hyperbola page to explore the geometric constructions of other conic sections.