FAD1014: Rapid-Fire Drill Pack — Hyperbola

Objective: Maximize problem volume in minimal time. Pure pattern recognition and mechanical fluency.
Target: 60–90 seconds per standard problem. If you stall >3 minutes, skip and mark it.
Total problems: 52
Estimated time: 2 hours (or split into four 30-minute sprints)

Cheat Sheet (Memorize First)

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 Formulas

Feature Horizontal Vertical
Centre $(h, k)$ $(h, k)$
Vertices $(h \pm a,; k)$ $(h,; k \pm a)$
Foci $(h \pm c,; k)$ $(h,; k \pm c)$
Asymptotes $y - k = \pm \frac{b}{a}(x - h)$ $y - k = \pm \frac{a}{b}(x - h)$
Latus rectum $\frac{2b^2}{a}$ $\frac{2b^2}{a}$
Relation $c^2 = a^2 + b^2$ $c^2 = a^2 + b^2$

Quick Identification

  • Positive $x^2$ term → opens left/right → Horizontal
  • Positive $y^2$ term → opens up/down → Vertical
  • Opposite signs in $x^2$ and $y^2$ terms → Hyperbola

Part A: Read Features from Standard Equation

Target: 45–60 seconds each. Extract centre, $a$, $b$, $c$, vertices, foci, asymptotes, and transverse axis.

Set A1 — Basic Horizontal Hyperbolas (6 problems)

For each equation, find: (i) centre, (ii) $a$ and $b$, (iii) $c$, (iv) vertices, (v) foci, (vi) asymptotes, (vii) length of transverse and conjugate axes.

  1. $\frac{x^2}{9} - \frac{y^2}{16} = 1$
  2. $\frac{(x-2)^2}{25} - \frac{(y+1)^2}{4} = 1$
  3. $\frac{(x+3)^2}{36} - \frac{(y-5)^2}{9} = 1$
  4. $4x^2 - y^2 = 16$ (Hint: divide to get 1 on RHS)
  5. $\frac{x^2}{1} - \frac{(y-4)^2}{12} = 1$
  6. $9(x-1)^2 - 4(y+2)^2 = 36$

Score: ___/6

Set A2 — Basic Vertical Hyperbolas (6 problems)

Same instructions as Set A1.

  1. $\frac{y^2}{25} - \frac{x^2}{16} = 1$
  2. $\frac{(y+2)^2}{9} - \frac{(x-1)^2}{4} = 1$
  3. $\frac{(y-3)^2}{49} - \frac{(x+4)^2}{24} = 1$
  4. $y^2 - 9x^2 = 9$
  5. $\frac{(y+1)^2}{5} - x^2 = 1$
  6. $16(y-2)^2 - 9(x+3)^2 = 144$

Score: ___/6

Set A3 — Latus Rectum & Eccentricity (6 problems)

For each equation, find the (i) length of latus rectum and (ii) eccentricity $e = \frac{c}{a}$.

  1. $\frac{x^2}{16} - \frac{y^2}{9} = 1$
  2. $\frac{y^2}{25} - \frac{x^2}{144} = 1$
  3. $\frac{(x+1)^2}{4} - \frac{(y-2)^2}{5} = 1$
  4. $\frac{(y-3)^2}{36} - \frac{(x+1)^2}{64} = 1$
  5. $25x^2 - 9y^2 = 225$
  6. $4y^2 - x^2 = 4$

Score: ___/6

Part B: Reverse Engineering — Find the Equation

Target: 60–90 seconds each. Use the given features to write the standard equation.

Set B1 — From Vertices and Foci (6 problems)

  1. Vertices at $(\pm 4, 0)$, foci at $(\pm 5, 0)$.
  2. Vertices at $(0, \pm 3)$, foci at $(0, \pm 5)$.
  3. Vertices at $(2, 6)$ and $(2, -2)$, foci at $(2, 7)$ and $(2, -3)$.
  4. Vertices at $(-1, 3)$ and $(5, 3)$, foci at $(-2, 3)$ and $(6, 3)$.
  5. Vertices at $(0, \pm 2)$, eccentricity $e = 3$.
  6. Foci at $(4, 0)$ and $(-4, 0)$, conjugate axis length $= 6$.

Score: ___/6

Set B2 — From Centre, Asymptotes, and One Feature (6 problems)

  1. Centre $(0, 0)$, asymptotes $y = \pm 2x$, vertex at $(3, 0)$.
  2. Centre $(0, 0)$, asymptotes $y = \pm \frac{3}{4}x$, vertex at $(0, 5)$.
  3. Centre $(2, -1)$, asymptotes $y + 1 = \pm \frac{2}{3}(x - 2)$, and passes through $(5, 1)$.
  4. Centre $(-3, 4)$, asymptotes $y - 4 = \pm (x + 3)$, and passes through $(-1, 6)$.
  5. Asymptotes $y = \pm \frac{1}{2}x$, passes through $(4, 3)$, horizontal transverse axis.
  6. Asymptotes $y = \pm 3x$, passes through $(2, 5)$, vertical transverse axis.

Score: ___/6

Set B3 — From General Conditions (4 problems)

  1. Difference of distances from any point to $(\pm 5, 0)$ is $8$.
  2. Difference of distances from any point to $(0, \pm 6)$ is $10$.
  3. Foci at $(-2, 5)$ and $(-2, -1)$, transverse axis length $= 4$.
  4. Vertices at $(1, -1)$ and $(1, 5)$, and passes through $(3, 4)$.

Score: ___/4

Part C: Completing the Square & General Form

Target: 90–120 seconds each. Convert to standard form, then list centre, $a$, $b$, and asymptotes.

  1. $x^2 - 4y^2 + 6x + 16y - 11 = 0$
  2. $9x^2 - 4y^2 - 36x - 24y - 36 = 0$
  3. $4y^2 - x^2 + 2x + 24y + 31 = 0$
  4. $16x^2 - 9y^2 + 32x + 54y - 79 = 0$
  5. $x^2 - y^2 - 4x + 6y - 6 = 0$ (Hint: this is a special case)
  6. $25y^2 - 4x^2 + 50y + 16x - 59 = 0$

Score: ___/6

Part D: Asymptotes, Graphing & Intersections

Target: 60–90 seconds each.

Set D1 — Asymptote Focus (4 problems)

  1. Find the asymptotes of $3x^2 - y^2 + 6x + 4y + 1 = 0$.
  2. Find the acute angle between the asymptotes of $\frac{x^2}{25} - \frac{y^2}{16} = 1$.
  3. Find the point where the hyperbola $\frac{(x-1)^2}{9} - \frac{(y+2)^2}{4} = 1$ intersects its horizontal transverse axis.
  4. Show that the asymptotes of $xy = k$ (rectangular hyperbola) are the coordinate axes.

Score: ___/4

Set D2 — Intersections with Lines (4 problems)

  1. Find the intersection points of $\frac{x^2}{9} - \frac{y^2}{4} = 1$ and $y = x - 1$.
  2. Find the intersection points of $\frac{y^2}{16} - \frac{x^2}{9} = 1$ and $2y - 3x = 0$.
  3. The line $y = 2x + c$ is tangent to $\frac{x^2}{4} - \frac{y^2}{9} = 1$. Find $c$.
  4. Find the equation of the tangent to $\frac{x^2}{16} - \frac{y^2}{9} = 1$ at the point $(5, \frac{9}{4})$.

Score: ___/4

Part E: Rapid Mixed Fire

Target: 60–90 seconds each. Mixed extraction, identification, and reverse problems.

  1. Identify the conic and find its centre: $2x^2 - 3y^2 + 4x + 12y - 8 = 0$.
  2. Find the equation of the hyperbola with vertices at $(\pm 2, 0)$ and asymptotes $y = \pm \frac{3}{2}x$.
  3. A hyperbola has eccentricity $\sqrt{5}$ and foci at $(0, \pm 10)$. Find its standard equation.
  4. Find $c$ such that $9x^2 - 16y^2 - 18x + 32y + c = 0$ represents a hyperbola with centre $(1, 1)$ and transverse axis length $8$.

Score: ___/4

Final Scorecard

Part Sets Problems Raw Score
A — Read Features from Equation A1, A2, A3 18 ___/18
B — Reverse Engineering B1, B2, B3 16 ___/16
C — Completing the Square — 6 ___/6
D — Asymptotes, Graphing & Intersections D1, D2 8 ___/8
E — Rapid Mixed Fire — 4 ___/4
TOTAL 52 ___/52

Proficiency Benchmarks

  • 38/52 (73%) — Proficient. You can handle standard exam problems.
  • 44/52 (85%) — Solid. Fast and accurate.
  • 49/52 (94%) — Exam-ready. Any mistake is a careless slip.

Speed Benchmarks

  • <1.5 hours: Excellent mechanical fluency.
  • 1.5–2 hours: Good. Review missed patterns.
  • >2.5 hours: Drill the specific sets you scored lowest on again tomorrow.

Error Log Template

After grading, list every wrong problem number with a one-word reason:

Problem Reason
e.g. 4 forgot to divide RHS
e.g. 21 mixed up $a$ and $c$

Re-solve all wrong problems immediately with notes, then again in 24 hours without notes.

Related Resources

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