Conjugate axis

Definition

The conjugate axis of a hyperbola is the line segment perpendicular to the transverse axis and passing through the center of the hyperbola. It connects two points called co-vertices.

5 Must Know Facts For Your Next Test

1. The length of the conjugate axis is $2b$, where $b$ is derived from the equation of the hyperbola.
2. The endpoints of the conjugate axis are located at $(h, k \pm b)$ for a horizontal hyperbola or $(h \pm b, k)$ for a vertical hyperbola.
3. The conjugate axis does not intersect the hyperbola itself; it lies entirely within the space between its branches.
4. In standard form, a horizontal hyperbola's equation is $\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1$, and $y$ varies with $b$ along the conjugate axis.
5. For a vertical hyperbola, its standard form is $\frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1$, and $x$ varies with $b$ along this axis.

Review Questions

• What is the relationship between the length of the conjugate axis and parameter $b$?
• How do you find the endpoints of the conjugate axis in both horizontal and vertical hyperbolas?
• Does the conjugate axis intersect any part of a hyperbola?

Related terms

Transverse Axis: The line segment that passes through both foci of a hyperbola and has endpoints on its vertices.

Vertices: The points where each branch of a hyperbola intersects its transverse axis.

$a$, $b$, and $c$: $a$ represents half-length of transverse axis, $b$ represents half-length of conjugate axis, and $c$ represents distance from center to each focus, with relation: $c^2 = a^2 + b^2$.