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Hyperbola Construction by Common Focal Distance in GeoGebra

Hyperbola Construction by Common Focal Distance in GeoGebra
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2015-03-28 04:08
Coordinate GeometryGeometric ConstructionSlider AnimationLocusGraphing
This GeoGebra example demonstrates the dynamic construction of a hyperbola based on its focal definition: the locus of points for which the absolute difference of distances to two fixed foci is constant. Core mathematical concepts include conic sections, hyperbolic focal properties, transverse axis, analytic geometry, distance formula, locus generation, and parametric control. The construction uses GeoGebra commands Circle, Line, Point, Intersect, and Segment, with objects F1 and F2 as foci, parameters n, p, and a, vertices, intersection points, auxiliary circles, segments, and text labels to determine and trace points on the hyperbola. Suitable for high school analytic geometry, pre-calculus, and undergraduate mathematics instruction, this resource supports classroom demonstrations, interactive exploration, and self-paced learning of hyperbolic curves.

GeoGebra Commands & Parameters

  1. Circle(F2, (n + Segment[F1, F2]) / 2) → p
  2. Line(F1, F2) → a
  3. Point(a) → G
  4. Intersect(p, a, 1) → Vertex
  5. Segment(Vertex, G) → i
  6. Point(i) → C
  7. Circle(F1, C) → c
  8. Segment(F1, C) → b
  9. Circle(F2, n + b) → d
  10. Intersect(c, d, 1) → D
  11. Segment(F1, D) → e
  12. Segment(D, F2) → f
  13. Intersect(c, d, 2) → E
  14. Circle(F1, f - n) → g
  15. Circle(F1, f - n) → h
  16. Circle(C, 0.5482549935540113) → k

AI Drawing Prompt Reference

Draw a hyperbola construction. Start with two fixed foci. Create an auxiliary circle centered at one focus with radius equal to half the distance between the foci. Draw a line through both foci, pick a point on it, and find the intersection of this line with the auxiliary circle as a vertex. Connect this vertex to the point on the line with a segment, and choose a point on that segment. Using this point as center, draw a circle with radius equal to its distance to the first focus. Also draw another circle centered at the second focus with radius equal to that distance plus the distance between the foci. The two circles intersect at two points. Connect the first focus to each intersection, and connect each intersection to the second focus, marking points on the hyperbola. Finally, add an extra auxiliary circle to complete the figure.
The prompt above was compiled from the original drawing commands and is for reference only.