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]

Related Concepts