Conjugate axis

Definition

The conjugate axis of a hyperbola is the line segment that passes through the center, perpendicular to the transverse axis, and has endpoints at the intersections with the hyperbola's asymptotes. Its length is equal to $2b$, where $b$ is one of the parameters defining the hyperbola.

5 Must Know Facts For Your Next Test

1. The conjugate axis is always perpendicular to the transverse axis.
2. Its length is $2b$, where $b$ comes from the hyperbola equation $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ or $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$.
3. It does not intersect the hyperbola itself.
4. For a hyperbola centered at $(h,k)$, its endpoints are at $(h, k \pm b)$ for horizontal orientation and $(h \pm b, k)$ for vertical orientation.
5. The conjugate axis helps in determining the asymptotes of the hyperbola.

Review Questions

• What is the length of the conjugate axis for a hyperbola given by $\frac{x^2}{9} - \frac{y^2}{16} = 1$?
• If a hyperbola has its center at $(3, -4)$ and $b=5$, what are the endpoints of its conjugate axis?
• How does the conjugate axis relate to finding the asymptotes of a hyperbola?

Related terms

Transverse Axis: The line segment that passes through both foci of a hyperbola and whose endpoints lie on the hyperbola itself. Its length is $2a$.

Asymptote: A line that a curve approaches as it heads towards infinity. For a hyperbola, there are two asymptotes that cross at its center.

$a$ (Semi-major Axis): $a$ represents half of the distance between vertices along the transverse axis in a standard form equation of a conic section such as an ellipse or hyperbola.