Ellipse Equations Demonstration
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① This example demonstrates the standard equation of an ellipse and its geometric definition, dynamically showing that the sum of distances from any point on the ellipse to the two foci is constant. ② Core mathematical concepts include the analytic‑geometry ellipse equation (x‑h)²/a²+(y‑k)²/b²=1, foci coordinates (h±c,k), major/minor axis lengths a, b, eccentricity e=c/a, and the focal‑distance property d₁+d₂=2a. ③ Construction highlights: GeoGebra commands If, Point, Segment are used to define parameters a, b, h, k, c, C; objects F₁, F₂, V₁, V₂, P are created, distances d₁, d₂ computed, and texts text1‑text4 label the equation and properties. ④ Suitable for high‑school analytic geometry (grades 10‑12) and as a visual aid in introductory college geometry courses.
GeoGebra Commands & Parameters
If(a ≠ b, If[a > b, (x(C) - f_1, y(C)), (x(C), y(C) + f_1)]) → F_1If(a ≠ b, If[a ≥ b, (x(C) + f_1, y(C)), (x(C), y(C) - f_1)]) → F_2Point(c) → PIf(a ≠ b, If[a ≥ b, (x(C) - a, y(C)), (x(C), y(C) + b)]) → V_1If(a ≠ b, If[a ≥ b, (x(C) + a, y(C)), (x(C), y(C) - b)]) → V_2Segment(If[a ≠ b, F_1, C], P) → d_1Segment(P, If[a ≠ b, F_2, C]) → d_2Segment(C, F_2) → fSegment(C, V_1) → d
AI Drawing Prompt Reference
Create a dynamic ellipse demonstration showing its standard equation and geometric definition. Start with the center, major and minor semi-axes, and determine the foci positions based on whether the major axis is horizontal or vertical. Place a moving point on the ellipse, draw segments from this point to each focus, and display that their sum is constant and equals the major axis length. Mark the two vertices, draw a segment from the center to a focus and another from the center to a vertex. Add text labels for the equation, focus coordinates, axis lengths, eccentricity, and the focal distance property.
The prompt above was compiled from the original drawing commands and is for reference only.