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Deriving a Parabola via Focus and Directrix Trace

Deriving a Parabola via Focus and Directrix Trace
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2016-06-09 22:13
Coordinate GeometryGeometric ConstructionLocus
This example demonstrates the dynamic derivation of a parabola's definition using the trace of points equidistant from a focus and a directrix. Key math concepts include parabola definition, focus, directrix, distance, angle, and coordinate geometry. GeoGebra commands used are Segment, Midpoint, OrthogonalLine, Intersect, Distance, Angle; objects include h, k, F, d, f, XQ, Q, i, C, j, l, P, FP, distanceFP, TextFP, distancePQ, TextPQ, α, β, distanceFC, TextFC, distanceCQ, TextCQ, text1, PQ, text2, text3. Suitable for high school coordinate geometry instruction or exploration by math enthusiasts.

GeoGebra Commands & Parameters

  1. Segment(F, Q) → i
  2. Midpoint(i) → C
  3. OrthogonalLine(C, i) → j
  4. OrthogonalLine(Q, f) → l
  5. Intersect(j, l) → P
  6. Segment(F, P) → FP
  7. Distance(F, P) → distanceFP
  8. Distance(P, Q) → distancePQ
  9. Angle(f, l) → α
  10. Angle(j, i) → β
  11. Distance(F, C) → distanceFC
  12. Distance(C, Q) → distanceCQ
  13. Segment(P, Q) → PQ

AI Drawing Prompt Reference

Draw a parabola with focus F and directrix d. First, draw a segment i through F and Q. Then, find the midpoint C of i and draw a line j through C that is perpendicular to i. Draw another line l through Q that is perpendicular to the directrix d, and find their intersection point P. Draw segment FP, calculate the lengths of FP and PQ, and the angles α and β between FP and PQ with the directrix d. Finally, calculate the distances from F to C and from C to Q.
The prompt above was compiled from the original drawing commands and is for reference only.