Hyperbola
A conic section defined as the set of all points where the absolute difference of distances to two foci is constant.
Standard Equations
| Orientation | Equation | Transverse axis |
|---|---|---|
| Horizontal | $\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1$ | Horizontal ($y = k$) |
| Vertical | $\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1$ | Vertical ($x = h$) |
Key Relations
- $a^2 + b^2 = c^2$ (where $c$ is focal distance from centre)
- Vertices are $a$ units from centre along transverse axis
- Foci are $c$ units from centre along transverse axis
Asymptotes
- Horizontal: $y - k = \pm \frac{b}{a}(x - h)$
- Vertical: $y - k = \pm \frac{a}{b}(x - h)$
Conic Section Relationships
flowchart LR
CONICS((Conic Sections))
CONICS --> ELLIPSE[Ellipse<br/>Sum of distances to foci is constant]
CONICS --> PARABOLA[Parabola<br/>Equidistant from focus and directrix]
CONICS --> HYPERBOLA[Hyperbola<br/>Difference of distances to foci is constant]
ELLIPSE --> CIRCLE[Circle<br/>Special case: a = b]
Identifying Conic Sections
flowchart TD
A["General Equation: Ax² + Cy² + Dx + Ey + F = 0"] --> B{"Are both A and C present?"}
B -->|"Yes, same sign"| C[Ellipse family]
B -->|"Yes, opposite signs"| D[Hyperbola]
B -->|"Only one present"| E[Parabola]
C --> F{"A = C?"}
F -->|"Yes"| G[Circle]
F -->|"No"| H[Ellipse]