Conic Sections Formulas
Standard equations for ellipses, parabolas, and hyperbolas.
Learn conic section formulas with key properties and worked examples.
Ellipse (Center at Origin)
An ellipse is a stretched circle. The values a and b are the semi-major and semi-minor axes (the longest and shortest radii). When a = b, the ellipse becomes a circle.
Key properties:
- Foci are located at (±c, 0) where c² = a² − b² (if a > b)
- Eccentricity: e = c/a (0 for a circle, approaches 1 as the ellipse gets thinner)
- The sum of distances from any point on the ellipse to both foci equals 2a
Parabola (Vertex at Origin)
A parabola is the set of all points equidistant from a point (focus) and a line (directrix). The vertex is at the turning point, and the axis of symmetry passes through the vertex and focus.
Key properties:
- Standard form with vertex at origin: y² = 4px (opens right, focus at (p, 0))
- Vertex form: y = a(x − h)² + k, where (h, k) is the vertex
- Focus distance from vertex: p = 1/(4a)
Hyperbola (Center at Origin)
A hyperbola consists of two separate curves (branches) that open away from each other. It is the set of all points where the difference of distances from two foci is constant.
Key properties:
- Foci at (±c, 0) where c² = a² + b²
- Asymptotes: y = ±(b/a)x
- Eccentricity: e = c/a (always greater than 1)
Variables
| Symbol | Meaning |
|---|---|
| a | Semi-major axis (ellipse) or semi-transverse axis (hyperbola) |
| b | Semi-minor axis (ellipse) or semi-conjugate axis (hyperbola) |
| c | Distance from center to focus |
| e | Eccentricity (shape parameter) |
| p | Distance from vertex to focus (parabola) |
Example 1: Ellipse
An ellipse has the equation x²/25 + y²/9 = 1. Find the foci, eccentricity, and the semi-major axis.
a² = 25, so a = 5 (semi-major axis along x-axis)
b² = 9, so b = 3 (semi-minor axis along y-axis)
c² = a² − b² = 25 − 9 = 16, so c = 4
Eccentricity: e = c/a = 4/5 = 0.8
Foci at (±4, 0), semi-major axis = 5, eccentricity = 0.8 (fairly elongated)
Example 2: Hyperbola
A hyperbola has the equation x²/16 − y²/9 = 1. Find the asymptotes and foci.
a² = 16, a = 4. b² = 9, b = 3
Asymptotes: y = ±(b/a)x = ±(3/4)x
c² = a² + b² = 16 + 9 = 25, c = 5
Foci at (±5, 0). Asymptotes: y = ±0.75x. The branches open left and right.
When to Use These
Conic sections appear throughout science, engineering, and nature.
- Planetary orbits are ellipses (Kepler's first law)
- Satellite dishes and headlight reflectors use parabolic shapes
- GPS navigation uses hyperbolic position finding
- Architecture uses all three curves in arches, domes, and bridges
- Projectile paths (in a uniform field) are parabolas