Introduction to Hyperbolas

Learn what hyperbolas are, their key features, and how they differ from other conic sections.

Advanced25 minLesson
Printable worksheet

Definition

A hyperbola is the set of all points (x,y)(x, y) in a plane such that the absolute difference of distances from two fixed points (called foci) is constant.
Unlike an ellipse where we add the distances, in a hyperbola we subtract them:
∣d1−d2∣=2a|d_1 - d_2| = 2a
A hyperbola has two separate curved branches that open either:
  • Horizontally (left and right), or
  • Vertically (up and down)
The standard form equations are:
Horizontal hyperbola (opens left/right):
x2a2−y2b2=1\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1
Vertical hyperbola (opens up/down):
y2a2−x2b2=1\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1
Key relationship: c2=a2+b2c^2 = a^2 + b^2 (note: this is different from ellipses!)

Try it now

Which equation represents a hyperbola?

Worked Examples

Identify the center, vertices, and foci of x225−y216=1\frac{x^2}{25} - \frac{y^2}{16} = 1
1

Identify the type

The x2x^2 term is positive, so this opens horizontally (left/right) → Horizontal hyperbola
2

Find the center

The equation is in standard form centered at the origin → Center: (0,0)(0, 0)
3

Find aa and bb

a2=25⇒a=5a^2 = 25 \Rightarrow a = 5, b2=16⇒b=4b^2 = 16 \Rightarrow b = 4 → a=5a = 5, b=4b = 4
4

Find the vertices

Vertices are at (±a,0)(\pm a, 0) → Vertices: (−5,0)(-5, 0) and (5,0)(5, 0)
5

Find cc using c2=a2+b2c^2 = a^2 + b^2

c2=25+16=41⇒c=41c^2 = 25 + 16 = 41 \Rightarrow c = \sqrt{41} → c=41≈6.4c = \sqrt{41} \approx 6.4
6

Find the foci

Foci are at (±c,0)(\pm c, 0) → Foci: (−41,0)(-\sqrt{41}, 0) and (41,0)(\sqrt{41}, 0)

Common Mistakes

Using c2=a2−b2c^2 = a^2 - b^2 (ellipse formula) instead of c2=a2+b2c^2 = a^2 + b^2

Why it's wrong: In ellipses, c<ac < a so we subtract. In hyperbolas, c>ac > a so we add.

Correct: For hyperbolas, always use c2=a2+b2c^2 = a^2 + b^2. The foci are always farther from the center than the vertices.

Confusing which variable gets a2a^2

Why it's wrong: In the equation, a2a^2 is always under the positive term. For horizontal hyperbolas, that is x2x^2; for vertical, it is y2y^2.

Correct: The positive term determines orientation. aa is always associated with the positive term.

Writing asymptotes as y=±abxy = \pm\frac{a}{b}x for horizontal hyperbolas

Why it's wrong: The asymptote formula depends on orientation: horizontal uses ba\frac{b}{a}, vertical uses ab\frac{a}{b}.

Correct: For horizontal: y=±baxy = \pm\frac{b}{a}x. For vertical: y=±abxy = \pm\frac{a}{b}x.

Thinking the branches connect at some point

Why it's wrong: Unlike ellipses, hyperbolas have two separate branches that never touch or connect.

Correct: The two branches extend infinitely toward the asymptotes but never cross them.

Watch a video explanation

The same topic explained by another teacher, if a video helps you more.

Hyperbolas Explained | Vertices, Foci, Asymptotes, and Equations

by Understand The Math

Watch on YouTube

Interactive Visual

Function graph

-10-50510-505xyF₁F₂
  • c = √(a² + b²) = 6.4
  • b/a = 0.8
Asymptotex²/a² − y²/b² = 1
  • Foci: (-6.4, 0), (6.4, 0)
  • Vertices: (-5, 0), (5, 0)
  • Asymptotes: y = 0.8x, y = -0.8x

Function graph

-10-50510-505xyF₁F₂
Asymptotey²/16 − x²/9 = 1
  • Foci: (0, -5), (0, 5)
  • Vertices: (0, -4), (0, 4)
  • Asymptotes: y = 1.333x, y = -1.333x

Interactive Sandbox

Interactive Grapher

-10-50510-10-50510xy

Tap or hover over the graph to read off values.

Try these examples:

y = 0.8*sqrt(x^2 - 25)

y = -0.8*sqrt(x^2 - 25)

Expression Calculator

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Practice Problems

15 problems
Problem 1 of 15
Easy
Which equation represents a hyperbola?

Why It Matters

Hyperbolas appear throughout science and engineering:
  • GPS Navigation: The intersection of hyperbolas from satellite signals determines your location
  • Sonic Booms: The shock wave from a supersonic jet forms a hyperbolic cone
  • Astronomy: Some comets follow hyperbolic paths around the Sun
  • Cooling Towers: Nuclear power plant cooling towers have a hyperbolic shape for structural strength
  • Radio Telescopes: Hyperbolic mirrors focus signals in reflecting telescopes
Understanding hyperbolas is essential for physics, engineering, and advanced mathematics!

Real World Applications

GPS and Navigation Systems

GPS uses hyperbolas to determine your location. Each pair of satellites creates a hyperbola of possible positions based on time differences.

Example:

If signal from satellite A arrives 0.001 seconds before satellite B, you are on a hyperbola where all points have this time difference.

1Try It Yourself

Two GPS stations are 100 km apart. A receiver detects a time difference indicating it is 30 km closer to station A.

What is the value of 2a2a (the constant difference) for this hyperbola?

Step 1: Write the mathematical expression

The constant difference is:

Cooling Tower Design

The hourglass shape of nuclear cooling towers is a hyperboloid of revolution. This shape provides maximum strength with minimum material.

Example:

The cross-section of a cooling tower follows the equation x2900−y22500=1\frac{x^2}{900} - \frac{y^2}{2500} = 1 where units are in meters.

2Try It Yourself

A cooling tower has a hyperbolic cross-section where a=30a = 30 m and b=50b = 50 m.

What is the minimum width (diameter) of the tower?

Step 1: Write the mathematical expression

The minimum width occurs at the vertices, so diameter = 2a2a:

Sonic Booms

When an aircraft exceeds the speed of sound, the shock wave forms a cone. The intersection with the ground is a hyperbola.

Example:

The sonic boom reaches all points on this hyperbola at the same instant, which is why you hear a sudden loud crack.

3Try It Yourself

A jet creates a sonic boom pattern described by x2400−y2300=1\frac{x^2}{400} - \frac{y^2}{300} = 1 (units in km).

Find the asymptote slope, which indicates how the boom spreads.

Step 1: Write the mathematical expression

For a horizontal hyperbola, asymptote slope = ba\frac{b}{a}:

Key Takeaways

  • 1A hyperbola is the set of points where the absolute difference of distances from two foci is constant: ∣d1−d2∣=2a|d_1 - d_2| = 2a
  • 2Horizontal hyperbola: x2a2−y2b2=1\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 (opens left/right)
  • 3Vertical hyperbola: y2a2−x2b2=1\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 (opens up/down)
  • 4Key relationship: c2=a2+b2c^2 = a^2 + b^2 (foci are farther from center than vertices)
  • 5Asymptotes guide the shape: horizontal uses y=±baxy = \pm\frac{b}{a}x, vertical uses y=±abxy = \pm\frac{a}{b}x
  • 6The two branches never connect and approach asymptotes but never touch them

Frequently Asked Questions

Look for a subtraction between squared terms: x2a2−y2b2=1\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 or y2a2−x2b2=1\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1. The minus sign between the terms (with result equal to 1) indicates a hyperbola.
Look for a subtraction between squared terms: x2a2−y2b2=1\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 or y2a2−x2b2=1\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1. The minus sign between the terms (with result equal to 1) indicates a hyperbola.
In a hyperbola, the foci are farther from the center than the vertices. Since cc (focal distance) is larger than aa (vertex distance), we need to add b2b^2 to get a larger value.
Asymptotes are lines that the hyperbola approaches but never touches. As the branches extend toward infinity, they get closer and closer to these diagonal lines. The asymptotes form an X through the center.
While they might look similar, a hyperbola is defined by the constant difference from two foci, has asymptotes, and both branches curve toward parallel lines. Two parabolas would not have these properties.

Glossary

Hyperbola
A conic section formed by points where the absolute difference of distances from two foci is constant
Foci
Two fixed points used to define the hyperbola; the constant ∣d1−d2∣=2a|d_1 - d_2| = 2a applies to all points
Vertices
The points where each branch is closest to the center; located at distance aa from center on the transverse axis
Transverse axis
The line segment connecting the two vertices, passing through both foci
Conjugate axis
The line segment perpendicular to the transverse axis, with length 2b2b
Asymptotes
Two diagonal lines that the branches of the hyperbola approach but never touch
Center
The midpoint between the two foci (and between the two vertices)
Today's challengeOne problem, three tries

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