Conic Sections Explorer
Pick a conic type and adjust the semi-axes. Watch foci, vertices, eccentricity, and directrices update in real-time.
Conic Type
Properties
| Type | — |
| Equation | — |
| Foci | — |
| Eccentricity | — |
| Vertices | — |
| Extra | — |
The Math Behind It
Conic Sections
- General second-degree equation: Ax² + Bxy + Cy² + Dx + Ey + F = 0
- Circle (e = 0): x² + y² = r² — all points equidistant from center
- Ellipse (0 < e < 1): x²/a² + y²/b² = 1
- Foci of ellipse: c = √(a² - b²) when a > b
- Eccentricity: e = c/a
Hyperbola & Parabola
- Hyperbola (e > 1): x²/a² - y²/b² = 1
- Foci of hyperbola: c = √(a² + b²)
- Asymptotes: y = ±(b/a)x
- Parabola (e = 1): y² = 4px
- Focus at (p, 0), directrix at x = -p
- Relationship: c² = a² ± b² (+ for hyperbola, - for ellipse)
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