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Grade 10 Math Lab: Proof & Generalization FREE
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Ellipses & Orbits
Learning Objectives
- Write equations of ellipses
- Find foci and vertices
- Apply to orbital mechanics
Key Concepts
Ellipse
x²/a² + y²/b² = 1 where a > b
Foci
Two special points: c² = a² - b²
Theory
**Standard form:** x²/a² + y²/b² = 1
- a: Semi-major axis (larger)
- b: Semi-minor axis (smaller)
- c: Distance to foci, c² = a² - b²
**Vertices:** (±a, 0) and (0, ±b)
**Foci:** (±c, 0) if horizontal
**Eccentricity:** e = c/a (0 < e < 1, closer to 0 = more circular)
Examples
Ellipse x²/25 + y²/9 = 1. Find foci.
Solution: a²=25, b²=9, c²=16, c=4. Foci: (±4, 0)