Minor Axis of a Hyperbola

Q: Can the minor axis of a hyperbola be longer than the major axis?

Yes. In a hyperbola, $b$ can be greater than $a$. The labels 'major' and 'minor' refer to which axis contains the vertices and foci (the transverse axis), not to which is longer. This is a key difference from ellipses, where the major axis is always the longer one.

Minor Axis of a Hyperbola

A line through the center of a hyperbola which is perpendicular to the major axis. The minor axis is an axis of symmetry.

Hyperbola with two curved branches opening left and right, with vertical "minor axis" arrow through center.

Key Formula

\text{Length of minor axis} = 2b

Where:

$b$ = The semi-minor axis length. For a hyperbola in standard form, $b$ appears in the denominator of the second fraction. It is related to $a$ (semi-major axis) and $c$ (distance from center to focus) by $c^2 = a^2 + b^2$.
$a$ = The semi-major (semi-transverse) axis length — the distance from the center to each vertex along the major axis.
$c$ = The distance from the center to each focus of the hyperbola.

Worked Example

Problem: Find the length of the minor axis for the hyperbola

\dfrac{x^2}{16} - \dfrac{y^2}{9} = 1

Step 1: Identify the standard form. This hyperbola opens left-right because the positive term is under

x^2

. The standard form is

\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1

a^2 = 16,\quad b^2 = 9

Step 2: Find

b

by taking the square root of

b^2

b = \sqrt{9} = 3

Step 3: The minor axis is the vertical segment through the center

(0,0)

, running from

(0, -b)

(0, b)

. Its length is

2b

\text{Length} = 2b = 2(3) = 6

Step 4: Identify the endpoints. The minor axis endpoints are

(0, -3)

and

(0, 3)

. Note: unlike an ellipse, the hyperbola does not actually pass through these points.

Answer: The minor axis has length 6, with endpoints at

(0, -3)

and

(0, 3)

Another Example

This example differs by using a vertical hyperbola with a shifted center, showing that $b$ can be larger than $a$ in a hyperbola (unlike an ellipse) and that you must account for the center coordinates when finding endpoints.

Problem: A hyperbola has the equation

\dfrac{(y-2)^2}{25} - \dfrac{(x+1)^2}{49} = 1

. Find the length of the minor axis and the coordinates of its endpoints.

Step 1: Identify the orientation. Because the positive term contains

y

, this hyperbola opens up and down. The center is at

(-1, 2)

a^2 = 25,\quad b^2 = 49

Step 2: Here

a^2 = 25

corresponds to the transverse (major) axis direction (vertical), and

b^2 = 49

corresponds to the conjugate (minor) axis direction (horizontal). Find

b

b = \sqrt{49} = 7

Step 3: Compute the length of the minor axis.

\text{Length} = 2b = 2(7) = 14

Step 4: Find the endpoints. The minor axis is horizontal through the center

(-1, 2)

, so move

b = 7

units left and right from the center.

(-1 - 7,\; 2) = (-8, 2) \quad\text{and}\quad (-1 + 7,\; 2) = (6, 2)

Answer: The minor axis has length 14, with endpoints at

(-8, 2)

and

(6, 2)

Frequently Asked Questions

What is the difference between the minor axis and the major axis of a hyperbola?

The major axis (also called the transverse axis) runs through the two vertices and the two foci of the hyperbola; its length is

2a

. The minor axis (also called the conjugate axis) is perpendicular to the major axis at the center; its length is

2b

. The hyperbola crosses the major axis at the vertices but does not cross the minor axis at all.

Does the hyperbola pass through the endpoints of the minor axis?

No. Unlike an ellipse, a hyperbola does not intersect its minor axis. The endpoints of the minor axis (called co-vertices) are geometric reference points used to construct the rectangle that defines the asymptotes, but the curve itself never reaches them.

Can the minor axis of a hyperbola be longer than the major axis?

Yes. In a hyperbola,

b

can be greater than

a

. The labels 'major' and 'minor' refer to which axis contains the vertices and foci (the transverse axis), not to which is longer. This is a key difference from ellipses, where the major axis is always the longer one.

Minor Axis (Conjugate Axis) vs. Major Axis (Transverse Axis)

	Minor Axis (Conjugate Axis)	Major Axis (Transverse Axis)
Definition	Segment through the center, perpendicular to the transverse axis	Segment through the center connecting the two vertices
Length	$2b$	$2a$
Contains foci?	No	Yes
Hyperbola crosses it?	No — the curve never intersects the minor axis	Yes — the curve crosses at both vertices
Always shorter?	Not necessarily; $b$ can exceed $a$	Not necessarily the longer axis
Role in asymptotes	Defines the $b$ -dimension of the asymptote rectangle	Defines the $a$ -dimension of the asymptote rectangle

Why It Matters

You encounter the minor axis of a hyperbola in precalculus and analytic geometry courses whenever you graph hyperbolas or derive their asymptotes. The value

b

from the minor axis determines the slopes of the asymptotes (

\pm\,b/a

for a horizontal hyperbola), which are essential for accurate sketching. In physics and engineering, hyperbolic shapes appear in satellite orbits, cooling tower profiles, and navigation systems (LORAN), where understanding both axes is necessary for precise modeling.

Common Mistakes

Mistake: Assuming the minor axis is always shorter than the major axis, as it is for ellipses.

Correction: For a hyperbola, the term 'minor' (conjugate) does not mean shorter. The value

b

can be larger than

a

. The major (transverse) axis is defined by which variable has the positive term in the standard equation, not by length.

Mistake: Thinking the hyperbola passes through the endpoints of the minor axis.

Correction: The hyperbola only crosses its transverse axis at the vertices. The endpoints of the minor axis are used to draw the reference rectangle and asymptotes, but the curve never touches them.

Related Terms

Hyperbola — The conic section that has this axis
Major Axis of a Hyperbola — Perpendicular axis through the vertices and foci
Axis of Symmetry — The minor axis is one of two symmetry axes
Perpendicular — Minor axis is perpendicular to major axis
Line — The minor axis lies along a line
Asymptote — Asymptote slopes depend on $b$ from the minor axis
Foci — Related by $c^2 = a^2 + b^2$
Vertex — Vertices lie on the major axis, not the minor axis