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"All-in-One" Loci: Ellipse, Hyperbola and Circle - Lesson

Author:Simona Riva
Move the foci F1and F2 that define the position of the focal axis of the conic, set the length of the major axis of the conic using the mAxis slider, then move point P along the circle to build the locus. To create the construction:
  • Plot 2 points, F1 and F2 that will be the foci of the conic.
  • Draw the circle with center F1 and radius r = mAxis
  • Create a point P on the circle, then draw the perpendicular bisector of segment PF2
  • Line PF1 intersects the perpendicular bisector at a point L, which is a point of the conic
Point P, moving on the circle, creates the locus of points of L, that is: - an ellipse if F2 is inside the circle distance between foci < axis length → e < 1 - an hyperbola if F2 is outside the circle   distance between foci > axis length → e > 1 - a circle if F1 ≡F2      distance between foci = 0 → e = 0 (e = eccentricity)

Explore the locus

After creating the locus, draw triangle PLF2 and answer the following questions:

What type of triangle is PLF2? Explain.

Write the canonical definition (as locus) of the conic that you see in the construction.

Use the properties of triangle PLF2 to show that the graph in the construction matches the canonical definition.

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