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Core Concepts — Ellipse
▼Standard Form (x-axis major)
x²/a² + y²/b² = 1, a > b > 0
Center (0,0). Major axis along x-axis. c² = a² − b², where c = focal distance.
Standard Form (y-axis major)
x²/b² + y²/a² = 1, a > b > 0
Center (0,0). Major axis along y-axis. Foci at (0, ±c).
Shifted Ellipse
(x−h)²/a² + (y−k)²/b² = 1
Center at (h, k). All distances measured from center.
Key Measurements
a = semi-major axis | b = semi-minor axis
c² = a² − b²
Eccentricity: e = c/a, 0 ≤ e < 1
Focal Chord & Latus Rectum
Latus rectum = 2b²/a
The chord through a focus perpendicular to the major axis.
Reflection Property
A ray from one focus reflects off the ellipse and passes through the other focus. Used in optics (whispering galleries, lithotripsy).
Directrix
x = ±a/e = ±a²/c
For every point P on the ellipse: PF/PD = e (focus-directrix property).
Parametric Form
x = a·cos t, y = b·sin t
t ∈ [0, 2π]. Useful for calculus and curve tracing.
💡 Quick example: For x²/25 + y²/9 = 1: a=5, b=3, c=√(25−9)=4. Foci: (±4, 0), vertices: (±5, 0), co-vertices: (0, ±3), e = 4/5 = 0.8.