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Conjugate axis

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Algebra and Trigonometry

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.

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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?
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