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8.2 The Hyperbola

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* Locate a hyperbolas vertices and foci. * Write equations of hyperbolas in standard form. * Graph hyperbolas centered at the origin. * Graph hyperbolas not centered at the origin. * Solve applied problems involving hyperbolas.

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8.2 The Hyperbola Chapter 8 Analytic Geometry
Concepts and Objectives The objectives for this section are Locate a hyperbolas vertices and foci. Write equations of hyperbolas in standard form. Graph hyperbolas centered at the origin. Graph hyperbolas not centered at the origin. Solve applied problems involving hyperbolas.
Hyperbolas Hyperbolas have two disconnected branches. Each branch approaches diagonal asymptotes. Parts of a hyperbola: Center Vertices Asymptotes Hyperbola
Hyperbolas The general equation of a hyperbola is or The hyperbola opens in whichever direction has the positive term (x-direction if x is positive, y-direction if y is positive). The slope of the asymptotes is always . The vertices are rx or ry from the center, whichever term is positive. a is the positive term radius, b is the negative term radius. = 2 2 1 x y x h y k r r + = 2 2 1 x y x h y k r r y x r r
Hyperbolas Example: Graph + + + = 2 2 9 4 90 32 197 0 x y x y
Hyperbolas Example: Graph + + + = 2 2 9 4 90 32 197 0 x y x y ( ) ( ) ( ) ( ) + + = + + 2 2 2 2 2 2 9 10 8 197 5 9 5 4 4 4 4 x x y y Notice that the negative sign has been factored as well! Remember to distribute the negative sign!
Hyperbolas Example: Graph + + + = 2 2 9 4 90 32 197 0 x y x y ( ) ( ) ( ) ( ) + + = + + 2 2 2 2 2 2 9 10 8 197 5 9 5 4 4 4 4 x x y y ( ) ( ) + = 2 2 9 6 4 4 5 3 x y Notice that the negative sign has been factored as well! Remember to distribute the negative sign!
Hyperbolas Example: Graph + + + = 2 2 9 4 90 32 197 0 x y x y ( ) ( ) ( ) ( ) + + = + + 2 2 2 2 2 2 9 10 8 197 5 9 5 4 4 4 4 x x y y ( ) ( ) + = 2 2 9 5 4 4 36 36 36 36 x y Notice that the negative sign has been factored as well! Remember to distribute the negative sign!
Hyperbolas Example: Graph + + + = 2 2 9 4 90 32 197 0 x y x y ( ) ( ) ( ) ( ) + + = + + 2 2 2 2 2 2 9 10 8 197 5 9 5 4 4 4 4 x x y y ( ) ( ) + = 2 2 9 5 4 4 36 36 36 36 x y ( ) ( ) + + = 2 2 5 4 1 4 9 x y Notice that the negative sign has been factored as well! Remember to distribute the negative sign!
Hyperbolas Example: Graph + + + = 2 2 9 4 90 32 197 0 x y x y ( ) ( ) ( ) ( ) + + = + + 2 2 2 2 2 2 9 10 8 197 5 9 5 4 4 4 4 x x y y ( ) ( ) + = 2 2 9 5 4 4 36 36 36 36 x y ( ) ( ) + + = 2 2 5 4 1 4 9 x y ( ) ( ) + + = 2 2 2 2 5 4 1 2 3 x y Notice that the negative sign has been factored as well! Remember to distribute the negative sign!
Hyperbolas Example: Graph Center (5, 4) + + + = 2 2 9 4 90 32 197 0 x y x y ( ) ( ) + + = 2 2 2 2 5 4 1 2 3 x y
Hyperbolas Example: Graph Center (5, 4) opens in y-direction rx = 2, ry = 3 vertices 3 + + + = 2 2 9 4 90 32 197 0 x y x y ( ) ( ) + + = 2 2 2 2 5 4 1 2 3 x y
Hyperbolas Example: Graph Center (5, 4) opens in y-direction rx = 2, ry = 3 vertices 3 slope of asymptotes: + + + = 2 2 9 4 90 32 197 0 x y x y ( ) ( ) + + = 2 2 2 2 5 4 1 2 3 x y 3 2
Hyperbolas Example: Graph Center (5, 4) opens in y-direction rx = 2, ry = 3 vertices 3 slope of asymptotes: + + + = 2 2 9 4 90 32 197 0 x y x y ( ) ( ) + + = 2 2 2 2 5 4 1 2 3 x y 3 2
Focal Length In an ellipse, the sum of the distances from a point on the ellipse to the two foci is constant, but in a hyperbola, its the difference between the distances that is constant. To find the focal radius, we can use the Pythagorean Theorem. Notice that c > a for the hyperbola. a b c = + 2 2 2 c a b
Classwork College Algebra 2e 8.2: 12-44 (4); 8.1: 32-56 (4); 7.8: 36-44 (4) 8.2 Classwork Check Quiz 8.1

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8.2 The Hyperbola

  • 1. 8.2 The Hyperbola Chapter 8 Analytic Geometry
  • 2. Concepts and Objectives The objectives for this section are Locate a hyperbolas vertices and foci. Write equations of hyperbolas in standard form. Graph hyperbolas centered at the origin. Graph hyperbolas not centered at the origin. Solve applied problems involving hyperbolas.
  • 3. Hyperbolas Hyperbolas have two disconnected branches. Each branch approaches diagonal asymptotes. Parts of a hyperbola: Center Vertices Asymptotes Hyperbola
  • 4. Hyperbolas The general equation of a hyperbola is or The hyperbola opens in whichever direction has the positive term (x-direction if x is positive, y-direction if y is positive). The slope of the asymptotes is always . The vertices are rx or ry from the center, whichever term is positive. a is the positive term radius, b is the negative term radius. = 2 2 1 x y x h y k r r + = 2 2 1 x y x h y k r r y x r r
  • 5. Hyperbolas Example: Graph + + + = 2 2 9 4 90 32 197 0 x y x y
  • 6. Hyperbolas Example: Graph + + + = 2 2 9 4 90 32 197 0 x y x y ( ) ( ) ( ) ( ) + + = + + 2 2 2 2 2 2 9 10 8 197 5 9 5 4 4 4 4 x x y y Notice that the negative sign has been factored as well! Remember to distribute the negative sign!
  • 7. Hyperbolas Example: Graph + + + = 2 2 9 4 90 32 197 0 x y x y ( ) ( ) ( ) ( ) + + = + + 2 2 2 2 2 2 9 10 8 197 5 9 5 4 4 4 4 x x y y ( ) ( ) + = 2 2 9 6 4 4 5 3 x y Notice that the negative sign has been factored as well! Remember to distribute the negative sign!
  • 8. Hyperbolas Example: Graph + + + = 2 2 9 4 90 32 197 0 x y x y ( ) ( ) ( ) ( ) + + = + + 2 2 2 2 2 2 9 10 8 197 5 9 5 4 4 4 4 x x y y ( ) ( ) + = 2 2 9 5 4 4 36 36 36 36 x y Notice that the negative sign has been factored as well! Remember to distribute the negative sign!
  • 9. Hyperbolas Example: Graph + + + = 2 2 9 4 90 32 197 0 x y x y ( ) ( ) ( ) ( ) + + = + + 2 2 2 2 2 2 9 10 8 197 5 9 5 4 4 4 4 x x y y ( ) ( ) + = 2 2 9 5 4 4 36 36 36 36 x y ( ) ( ) + + = 2 2 5 4 1 4 9 x y Notice that the negative sign has been factored as well! Remember to distribute the negative sign!
  • 10. Hyperbolas Example: Graph + + + = 2 2 9 4 90 32 197 0 x y x y ( ) ( ) ( ) ( ) + + = + + 2 2 2 2 2 2 9 10 8 197 5 9 5 4 4 4 4 x x y y ( ) ( ) + = 2 2 9 5 4 4 36 36 36 36 x y ( ) ( ) + + = 2 2 5 4 1 4 9 x y ( ) ( ) + + = 2 2 2 2 5 4 1 2 3 x y Notice that the negative sign has been factored as well! Remember to distribute the negative sign!
  • 11. Hyperbolas Example: Graph Center (5, 4) + + + = 2 2 9 4 90 32 197 0 x y x y ( ) ( ) + + = 2 2 2 2 5 4 1 2 3 x y
  • 12. Hyperbolas Example: Graph Center (5, 4) opens in y-direction rx = 2, ry = 3 vertices 3 + + + = 2 2 9 4 90 32 197 0 x y x y ( ) ( ) + + = 2 2 2 2 5 4 1 2 3 x y
  • 13. Hyperbolas Example: Graph Center (5, 4) opens in y-direction rx = 2, ry = 3 vertices 3 slope of asymptotes: + + + = 2 2 9 4 90 32 197 0 x y x y ( ) ( ) + + = 2 2 2 2 5 4 1 2 3 x y 3 2
  • 14. Hyperbolas Example: Graph Center (5, 4) opens in y-direction rx = 2, ry = 3 vertices 3 slope of asymptotes: + + + = 2 2 9 4 90 32 197 0 x y x y ( ) ( ) + + = 2 2 2 2 5 4 1 2 3 x y 3 2
  • 15. Focal Length In an ellipse, the sum of the distances from a point on the ellipse to the two foci is constant, but in a hyperbola, its the difference between the distances that is constant. To find the focal radius, we can use the Pythagorean Theorem. Notice that c > a for the hyperbola. a b c = + 2 2 2 c a b
  • 16. Classwork College Algebra 2e 8.2: 12-44 (4); 8.1: 32-56 (4); 7.8: 36-44 (4) 8.2 Classwork Check Quiz 8.1