FAD1014 Fast Revision — Hyperbola Mechanics

Goal: Perform hyperbola analysis on autopilot. No thinking — just execute.

1. Formula Card (Memorise Cold)

Mnemonic for asymptotes:

  • Horizontal hyperbola → slope is $\frac{b}{a}$ ("big horizontal steps")
  • Vertical hyperbola → slope is $\frac{a}{b}$ ("always vertical first")

2. Identification Protocol (2 Seconds)

Look at the standard equation:

  • Positive $x^2$ term → opens left/right → Horizontal
  • Positive $y^2$ term → opens up/down → Vertical

If the equation is not standard, complete the square first.

3. Mechanical Drill Template

Given any standard-form hyperbola, fill this in without hesitation.

1. Read off centre: (h, k) = (____, ____)
2. Read off a² and b²:
   a² = ____  →  a = ____
   b² = ____  →  b = ____
3. Compute c:
   c² = a² + b² = ____  →  c = ____
4. State orientation: Horizontal / Vertical
5. List vertices: ____________________
6. List foci:   ____________________
7. Write asymptotes:
   ____________________
8. Sketch: box 2a × 2b, draw diagonals = asymptotes, trace branches.

4. Reverse Engineering (Given Features → Equation)

Archetype A: Given vertices and foci

  1. Find centre = midpoint of vertices.
  2. Find $a$ = distance from centre to vertex.
  3. Find $c$ = distance from centre to focus.
  4. Compute $b^2 = c^2 - a^2$.
  5. Pick the correct standard equation based on whether vertices/foci lie horizontally or vertically.

Archetype B: Given asymptotes and a point

  1. Read the ratio $\frac{b}{a}$ (or $\frac{a}{b}$) from asymptote slopes.
  2. Assume a standard equation with unknown $a, b$.
  3. Plug in the given point; solve for $a^2$ and $b^2$.
  4. If only one equation exists, you may need an extra condition (e.g., vertex or focus) to fix both.

Archetype C: Given vertices and asymptotes

  1. Get $a$ and the ratio $\frac{b}{a}$ directly.
  2. Compute $b$.
  3. Write equation.

5. Common Exam Traps

Trap Fix
Forgetting $c^2 = a^2 + b^2$ (not $c^2 = a^2 - b^2$ like ellipse) Hyperbola adds; ellipse subtracts
Mixing up $a$ and $b$ in asymptotes Horizontal → $\frac{b}{a}$; Vertical → $\frac{a}{b}$
Wrong sign when completing the square Factor carefully; keep RHS = 1
Sketching branches touching the box Branches approach asymptotes, they never touch the box corners
Confusing transverse vs conjugate axis Transverse = $2a$ (through foci); Conjugate = $2b$

6. 60-Second Self-Check

Cover the table above and write from memory:

  1. Standard equation (horizontal) → ________
  2. Vertices for vertical → ________
  3. Asymptotes for horizontal → ________
  4. $c$ relation → ________
  5. Latus rectum length → ________

Pass criteria: All correct in under 60 seconds.

Links