"All-in-one" loci: ellipse, hyperbola and circle

Author:
Simona Riva
Topic:
Circle, Ellipse, Hyperbola
Move the foci F1 and 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 view the locus. To reproduce the construction:
  • 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 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.
  • What type of triangle is it? 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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