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Conjugate axis

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College Algebra

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.

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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?
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