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Ellipse Equation: Foci, Semi-Axes, and Geometric Construction

Ellipse Equation: Foci, Semi-Axes, and Geometric Construction
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2023-03-17 00:05
FunctionCoordinate GeometryGeometric ConstructionSlider AnimationLocus
In collections geometriadinamica.es
This GeoGebra example demonstrates the standard equation of an ellipse and its geometric construction, interactively showing that for any point P on the ellipse, the sum of distances to the two foci F and F' equals the major axis length 2a. Core mathematical concepts include the definition of an ellipse (locus of points with constant sum of distances to two foci), semi-major axis a, semi-minor axis b, focal distance c, and the relation c² = a² - b². Construction details: using GeoGebra commands Intersect, Point, Ray, and Segment, the construction includes center O, vertices A and B, foci F and F', semi-axis variables semi_a and semi_b, and a moving point P on the ellipse; segments PF and PF' visually illustrate the constant distance sum. Auxiliary circles cb, Fb, F'b and intersection points T1-T4 are also constructed to verify ellipse properties. Suitable for high school analytic geometry or introductory college geometry courses, this example helps students connect the algebraic equation with the geometric definition, and allows dynamic exploration by dragging sliders to change semi-axis lengths and observe the resulting shape changes.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → O
  2. Point(xAxis) → AA
  3. Ray(O, AA) → semi_a
  4. Point(semi_a) → A
  5. Point(yAxis) → BB
  6. Ray(O, BB) → semi_b
  7. Point(semi_b) → B
  8. Point(d) → P
  9. Segment(F', P) → PF'
  10. Segment(P, F) → PF
  11. Segment(Fb, P) → e
  12. Segment(P, F'b) → f

AI Drawing Prompt Reference

Draw an ellipse centered at the origin with foci on the x-axis. Use sliders to adjust the lengths of the semi-major and semi-minor axes and observe changes in the ellipse shape. First, draw the intersection of the x-axis and y-axis as the center O, then draw rays along the x-axis and y-axis representing the semi-major and semi-minor axes. Next, take points A and B on these rays as the ellipse's vertices. Finally, place a moving point P on the ellipse and draw line segments PF and PF' to the foci, and include auxiliary circles and intersection points to verify the ellipse properties.
The prompt above was compiled from the original drawing commands and is for reference only.