Conic Sections: Hyperbola

Definition

A hyperbola is the collection of all points $P$ in a plane, the absolute difference of whose distances from two fixed points $F_1$ and $F_2$ (the foci) is a constant. Mathematically, it is defined by the locus of points satisfying $|d(F_1, P) - d(F_2, P)| = 2a$, where $2a$ is the distance between the vertices.

Why It Matters

This geometric shape describes the paths of celestial bodies on escape trajectories and the “sonic boom” of supersonic aircraft. Its unique properties are essential for understanding gravity assists in space navigation and the physics of wave propagation.

Core Concepts

• Geometric definition (difference of distances)

  • $|d(F_1, P) - d(F_2, P)| = 2a$.
  • How to read: “The absolute value of the difference between the distance from F one to P and the distance from F two to P is equal to two a.”
  • Meaning: The absolute difference of distances from any point on the hyperbola to the two foci is the constant $2a$.

• Standard form — Horizontal transverse axis (opens left/right) $\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1$

  • How to read: “The quantity x minus h squared, divided by a squared, minus the quantity y minus k squared, divided by b squared, is equal to one.”
  • Meaning: Center $(h,k)$; transverse axis horizontal ($a$ along $x$); opens left/right; asymptotes $y - k = \pm (b/a)(x - h)$.

• Standard form — Vertical transverse axis (opens up/down) $\frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1$

  • How to read: “The quantity y minus k squared, divided by a squared, minus the quantity x minus h squared, divided by b squared, is equal to one.”
  • Meaning: Center $(h,k)$; transverse axis vertical ($a$ along $y$); opens up/down; asymptotes $y - k = \pm (a/b)(x - h)$.

• Key relationships (note the sign change vs ellipse)

  • c² = a² + b² (always c > a)
    • How to read: “The value c squared is equal to a squared plus b squared.”
    • Meaning: Foci are farther out than the vertices ($c > a$); unlike the ellipse, $c^2 = a^2 + b^2$ (not minus).
  • Eccentricity e = c / a > 1 (always greater than 1 for hyperbolas).
  • a = semi-transverse, b = semi-conjugate. The asymptotes have slopes ±b/a (or ±a/b).

• Decision rule: The positive term tells you the direction the hyperbola opens. The asymptotes always pass through the center and have slopes determined by b/a.

Connected Concepts

• Conic Sections: Ellipse
• Conic Sections: Parabola
• Conic Sections
• Rotation of Axes
• Trade-offs
• Leverage
• Map-Territory Relation
• Symmetry
• Scale
• Bottlenecks
• First Principles Thinking