Mathematics.

conic sections

Hyperbola

Analytic Geometry20 minDifficulty5 out of 10

You should know: conic sections

Overview

A hyperbola is a smooth plane curve with two mirror-image branches that resemble two infinite bows. It is one of the three conic sections, formed when a plane intersects both nappes of a double cone without passing through the apex. Algebraically, a hyperbola is the set of points where the difference of distances to two fixed foci is constant.

Formal Definition

Definition

A hyperbola centered at the origin with foci on the x-axis, in standard position:

x2a2−y2b2=1\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1

a = semi-transverse axis, b = semi-conjugate axis

Standard form
c2=a2+b2c^2 = a^2 + b^2

Foci located at (±c, 0)

Focal distance
H={P∈R2:∣∣PF1∣−∣PF2∣∣=2a}H = \{P \in \mathbb{R}^2 : \big| |PF_1| - |PF_2| \big| = 2a\}

Difference-of-distances definition using the two foci F₁, F₂

e=ca>1e = \frac{c}{a} > 1
Eccentricity
y=±baxy = \pm\frac{b}{a}x

Equations of the two asymptotes

Properties

Vertices

(±a,0), the closest points of each branch to the center(\pm a, 0), \text{ the closest points of each branch to the center}

Foci

(±c,0),c=a2+b2(\pm c, 0), \quad c = \sqrt{a^2+b^2}

Eccentricity

e=c/a>1; larger e means wider, flatter branchese = c/a > 1; \text{ larger } e \text{ means wider, flatter branches}

Asymptotes

y=±bax, the lines the branches approach at infinityy = \pm\frac{b}{a}x, \text{ the lines the branches approach at infinity}

Worked Examples

  1. Identify a² = 16, b² = 9, so a = 4, b = 3.

    a=4, b=3a=4,\ b=3

  2. Compute c using c² = a² + b².

    c2=16+9=25  ⟹  c=5c^2 = 16 + 9 = 25 \implies c = 5

  3. Compute eccentricity and asymptotes.

    e=5/4;y=±34xe = 5/4;\quad y = \pm\frac{3}{4}x

Answer: Foci at (±5, 0); eccentricity e = 5/4; asymptotes y = ±(3/4)x.

Practice Problems

Difficulty 5/10

Find a² and b² for a hyperbola with vertices (±6, 0) and foci (±10, 0).

Common Mistakes

Common Mistake

Using c² = a² - b² (the ellipse relation) instead of c² = a² + b² for a hyperbola.

For a hyperbola the foci lie outside the vertices, so c > a always, requiring c² = a² + b² (the '+' relation). The ellipse's '−' relation would give an imaginary or invalid c here.

Common Mistake

Forgetting that a hyperbola has two disconnected branches, not one continuous curve.

Unlike an ellipse or parabola, a hyperbola's graph consists of two separate curves opening in opposite directions, approaching but never touching the asymptotes.

Summary

  • A hyperbola is the set of points whose distances to two fixed foci differ by a constant, 2a.
  • Standard form x²/a² − y²/b² = 1; foci at (±c, 0) where c² = a² + b² (note the plus sign, unlike the ellipse).
  • Eccentricity e = c/a is always greater than 1.
  • The two branches approach the asymptotes y = ±(b/a)x but never touch them.

References

  1. WebsiteWikipedia — Hyperbola