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Major Axis of a Hyperbola — Definition, Formula & Examples

Major Axis of a Hyperbola

The line passing through the foci, center, and vertices of a hyperbola. It is also the principle axis of symmetry.

 

Hyperbola with two curves opening left and right, connected by a horizontal "major axis" line through the center and vertices.

 

 

See also

Minor axis of a hyperbola

Key Formula

Length of major axis=2a\text{Length of major axis} = 2a
Where:
  • aa = The distance from the center of the hyperbola to either vertex. In the standard form equation, $a^2$ appears under the positive term.
  • 2a2a = The total length of the major axis, measured from one vertex to the other.

Worked Example

Problem: Find the length and direction of the major axis for the hyperbola given by x225−y29=1.\frac{x^2}{25} - \frac{y^2}{9} = 1.
Step 1: Identify the standard form. The positive term is under x2x^2, so this is a horizontal hyperbola. The major axis (also called the transverse axis) lies along the xx-axis.
x2a2−y2b2=1⇒a2=25,  b2=9\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \quad \Rightarrow \quad a^2 = 25,\; b^2 = 9
Step 2: Solve for aa.
a=25=5a = \sqrt{25} = 5
Step 3: Find the vertices. Since the center is at the origin and the major axis is horizontal, the vertices are at (±a,0)(\pm a, 0).
Vertices: (−5, 0) and (5, 0)\text{Vertices: } (-5,\, 0) \text{ and } (5,\, 0)
Step 4: Calculate the length of the major axis.
Length=2a=2(5)=10\text{Length} = 2a = 2(5) = 10
Answer: The major axis is horizontal (along the xx-axis), and its length is 10 units.

Another Example

This example differs from the first by featuring a vertical hyperbola with a center not at the origin, requiring you to apply a shift to find the vertex coordinates.

Problem: Find the endpoints, length, and direction of the major axis for the hyperbola (y−2)216−(x+3)236=1.\frac{(y - 2)^2}{16} - \frac{(x + 3)^2}{36} = 1.
Step 1: Identify center and orientation. The positive term is under (y−2)2(y - 2)^2, so this is a vertical hyperbola. The center is at (−3,2)(-3, 2).
Center: (h,k)=(−3, 2)\text{Center: } (h, k) = (-3,\, 2)
Step 2: Read off a2a^2 from the positive term and find aa. For a vertical hyperbola, a2a^2 is the denominator under the yy-term.
a2=16⇒a=4a^2 = 16 \quad \Rightarrow \quad a = 4
Step 3: Find the vertices. Since the major axis is vertical, move aa units up and down from the center.
Vertices: (−3, 2+4)=(−3, 6) and (−3, 2−4)=(−3, −2)\text{Vertices: } (-3,\, 2 + 4) = (-3,\, 6) \text{ and } (-3,\, 2 - 4) = (-3,\, -2)
Step 4: Calculate the major axis length.
Length=2a=2(4)=8\text{Length} = 2a = 2(4) = 8
Answer: The major axis is vertical, runs from (−3,−2)(-3, -2) to (−3,6)(-3, 6), and has a length of 8 units.

Frequently Asked Questions

What is the difference between the major axis and the transverse axis of a hyperbola?
They refer to the same thing. The terms 'major axis' and 'transverse axis' are used interchangeably for a hyperbola — both describe the axis that passes through the two vertices and the two foci. Some textbooks prefer 'transverse axis' to avoid confusion with the major axis of an ellipse.
How do you tell if the major axis of a hyperbola is horizontal or vertical?
Look at which variable's term is positive in the standard form equation. If the x2x^2 term is positive (i.e., x2a2−y2b2=1\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1), the major axis is horizontal. If the y2y^2 term is positive (i.e., y2a2−x2b2=1\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1), the major axis is vertical.
Does the major axis of a hyperbola always pass through the foci?
Yes. By definition, the foci always lie on the major axis. The foci are located at a distance c=a2+b2c = \sqrt{a^2 + b^2} from the center, along the major axis. Since c>ac > a, the foci lie outside the segment connecting the two vertices but still on the same line.

Major Axis of a Hyperbola vs. Minor Axis of a Hyperbola

Major Axis of a HyperbolaMinor Axis of a Hyperbola
DefinitionThe axis passing through the center, vertices, and fociThe axis passing through the center, perpendicular to the major axis
Length formula2a2a (vertex-to-vertex distance)2b2b (co-vertex to co-vertex distance)
Intersects the curve?Yes — it intersects the hyperbola at both verticesNo — the hyperbola does not cross the minor axis
Contains foci?YesNo
Also calledTransverse axisConjugate axis

Why It Matters

You encounter the major axis whenever you graph a hyperbola, because it tells you the direction the curve opens and where the vertices sit. It is essential for computing the foci (c=a2+b2c = \sqrt{a^2 + b^2}) and for writing equations of asymptotes. In physics and astronomy, hyperbolic trajectories (such as those of comets or spacecraft performing gravity assists) are analyzed with respect to their major axis to determine the path's orientation.

Common Mistakes

Mistake: Confusing aa and bb based on which denominator is larger.
Correction: Unlike an ellipse, the value of a2a^2 in a hyperbola is always the denominator under the positive term — not necessarily the larger number. In a hyperbola, bb can be greater than aa.
Mistake: Using c=a2−b2c = \sqrt{a^2 - b^2} (the ellipse formula) instead of c=a2+b2c = \sqrt{a^2 + b^2} for finding the foci along the major axis.
Correction: For a hyperbola, the relationship is c2=a2+b2c^2 = a^2 + b^2. The foci are farther from the center than the vertices, so c>ac > a always holds.

Related Terms