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Focus-Directrix Definition of an Ellipse

Focus-Directrix Definition of an Ellipse
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2017-02-22 03:56
FunctionCoordinate GeometryGeometric ConstructionSlider AnimationLocus
This GeoGebra example dynamically illustrates the focus-directrix definition of an ellipse: an ellipse is the locus of points whose distance to a fixed point (focus F) and a fixed line (directrix) have a constant ratio (eccentricity e, 0<e<1). Key mathematical concepts include conic sections, focus, directrix, eccentricity, and locus generation. Construction uses commands such as Point, Sequence, Segment, OrthogonalLine, and Intersect, with sliders to control eccentricity, tracing point P to form the ellipse. Suitable for high school analytic geometry or introductory college geometry, helping students grasp the equivalence of ellipse definitions.

GeoGebra Commands & Parameters

  1. Point(λ) → B
  2. Sequence((((4 * sqrt(1 - ε^(2))) * cos(t)), (4 * ε) + (4 * sin(t))), t, 0, (2 * π), π / 36) → list1
  3. Point(list1) → P
  4. Segment(F, P) → f
  5. OrthogonalLine(P, λ) → g
  6. Intersect(λ, g) → N
  7. Segment(P, N) → h

AI Drawing Prompt Reference

Draw an ellipse to illustrate its focus-directrix definition: the locus of points where the ratio of distances to a fixed point (focus) and a fixed line (directrix) is constant (eccentricity, 0<eccentricity<1). Use a slider to control the eccentricity, trace a point to form the ellipse, and show the line segment from the focus to the point and the perpendicular from the point to the directrix.
The prompt above was compiled from the original drawing commands and is for reference only.