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Exploring Hyperbolas: Geometry, Translations, and Applications

Dive into the world of hyperbolas, learning about their geometry, translations, eccentricity, orbits, reflective property, and use in long-range navigation. Discover key points and equations to enhance your understanding.

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Exploring Hyperbolas: Geometry, Translations, and Applications

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  1. 8.3 Demana, Waits, Foley, Kennedy Hyperbolas

  2. What you’ll learn about • Geometry of a Hyperbola • Translations of Hyperbolas • Eccentricity and Orbits • Reflective Property of a Hyperbola • Long-Range Navigation … and why The hyperbola is the least known conic section, yet it is used astronomy, optics, and navigation.

  3. Hyperbola

  4. Hyperbola A hyperbola is the set of all points in a plane whose distances from two fixed points in the plane have a constant difference. The fixed points are the foci of the hyperbola. The line through the foci is the focal (or Transverse) axis. The point on the focal axis midway between the foci is the center. The points where the hyperbola intersects its focal axis are the vertices of the hyperbola.

  5. Hyperbola

  6. Let’s graph one

  7. Hyperbola with Center (0,0)

  8. Hyperbola Centered at (0,0)

  9. Example: Finding the Vertices and Foci of a Hyperbola

  10. Solution

  11. Example: Finding an Equation of a Hyperbola

  12. Solution

  13. Hyperbola with Center (h,k)

  14. Hyperbola with Center (h,k)

  15. Example: Finding an Equation of a Hyperbola

  16. Solution

  17. Solution Continued

  18. Example: Locating Key Points of a Hyperbola

  19. Solution

  20. Eccentricity of a Hyperbola

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