L31-L32: Geometry — Hyperbola

Lecture notes covering the definition, derivation, standard equation, and asymptotes of hyperbolas.

Key Points

  • A hyperbola is the set of all points where the absolute difference of distances from two fixed foci is constant ($\pm 2a$).
  • Two separate branches opening horizontally or vertically.
  • Transverse axis: passes through foci and vertices (length $2a$).
  • Conjugate axis: perpendicular to transverse axis at centre (length $2b$).
  • Relation: $a^2 + b^2 = c^2$ (where $c$ is focal distance from centre).
  • Latus rectum: perpendicular to transverse axis through a focus.

Standard Equations

Horizontal orientation (centre $(h,k)$): $$ \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 $$

Vertical orientation (centre $(h,k)$): $$ \frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1 $$

Asymptotes

From the horizontal standard equation: $$ y - k = \pm \frac{b}{a}(x - h) $$

Vertical orientation: $$ y - k = \pm \frac{a}{b}(x - h) $$

Links