Geometría interactiva y funciones · AP Precalculus Unit 4 (Conic Sections & Polar Coordinates), A-Level Further Maths (Conics), High School Analytic Geometry.
Visualizador de secciones cónicas y elipses
Visualizador geométrico de secciones cónicas y construcción de elipses: focos y excentricidad
- Cálculo en cliente
- Sin registro
- Derivación en vivo
- Modo proyector
- Hoja de trabajo A4
Ventana de demostración geométrica● 60 FPS
Variables de control del experimento
cm
cm
°
°
modo
Derivaciones y métricas en tiempo real
Semieje menor b
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b = √(a² - c²)
Excentricidad e
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e = c / a (0 ≤ e < 1)
Longitud total del hilo 2a
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Invariant PF1 + PF2 = 2a
Distancia entre clavos (focos) 2c
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|F1F2| = 2c
Derivación de fórmulas en tiempo real
Fórmula básica:
--Sustitución de valores:
--Conclusión: --
Ejemplos clásicos y análisis de errores en exámenes
Analytic Geometry: Ellipse with a=5, c=4, Find Standard Equation & Directrix
Problema: An ellipse has foci on the x-axis with semi-major axis a = 5 and focal half-distance c = 4. Find its standard equation, eccentricity e, and directrix equations.
Estrategia analítica: From a² = b² + c², we get b² = 25 - 16 = 9 (b=3). Standard equation is x²/25 + y²/9 = 1. Eccentricity e = c/a = 4/5 = 0.8. Directrices are x = ±a²/c = ±25/4.
Derivación paso a paso:
- Step 1: Compute semi-minor axis b: b = √(a² - c²) = √(25 - 16) = 3.
- Step 2: Foci on x-axis => equation is x²/25 + y²/9 = 1.
- Step 3: Compute eccentricity e = c / a = 4 / 5 = 0.8.
- Step 4: Compute directrices: x = ±a²/c = ±25/4 = ±6.25.
⚠️ Error conceptual frecuente: Confusing conic relationships: for an ellipse, a is largest (a² = b² + c²); for a hyperbola, c is largest (c² = a² + b²).
Hoja de indagación en clase y tareas
Q1. Two pins are spaced 6 cm apart holding a 10 cm string taut. What is the semi-minor axis b?
Answer Key: 4 cm (2a=10=>a=5, 2c=6=>c=3, b=√(5²-3²)=4)
Q2. Which physical property of an ellipse is utilized in medical lithotripters for kidney stone treatment?
Answer Key: Acoustic reflection focal property (waves emitted from one focus reflect and concentrate at the other focus)
Preguntas frecuentes y errores conceptuales
How do slice angle β and cone generator angle α dictate the conic section type?
Comparing slice angle β to cone generator slope α: β=0 gives a circle; β < α gives an ellipse; β = α (parallel to generator) produces a parabola; β > α (cutting both nappes) produces a hyperbola.