1. OHHS Pre-Calculus Mr. J. Focht

2. 8.3 Hyperbolas Geometry of a Hyperbola Translations of Hyperbolas Eccentricity 8.3

3. A Hyperbola is a Conic Section 8.3

4. Hyperbola Definition Set of all points whose difference of the distances to 2 fixed points is constant. Focus (x,y) d1d1 d2d2 d 1 – d 2 is the same for whatever (x,y) you choose on the blue curves. 8.3

5. Hyperbola Terms Focus Center Transverse Axis Conjugate Axis Vertex Asymptote Focal Axis 8.3

6. Hyperbola Terms Focus Center Asymptote a = distance from center to vertex a b b = distance from vertex to asymptote c = distance from center to focus c c 2 = a 2 + b 2 8.3

7. Hyperbola Equation Focus Center Asymptote a b c (h,k) 8.3

8. Hyperbola Equation Focus Asymptote a b c (h,k) 8.3

9. Asymptote Equations Focus Center Asymptote a b c (h,k) 8.3

10. Asymptote Equations Focus Asymptote a b c (h,k) 8.3

11. Example Find the vertices and the foci of the hyperbola 4x 2 - 9y 2 = 36. Divide by sides by 36. 8.3

12. Example: Find the Center, Vertices, and Foci a 2 = 9 a = 3 Vertices (-3, 0), (3, 0) (h, k) = (0, 0) b 2 = 4 c 2 = a 2 + b 2 = 9 + 4 = 13 Foci 8.3

13. Now You Try Find the center, vertices, and foci 8.3

14. Find the equation of the hyperbola 4 (1,-5) (1, 1) The center is halfway between the foci. (1, -2) 2 2 a 1 +2 4 c = distance from center to a focus c = 3 c 2 = a 2 + b 2 9 = 4 + b 2 b 2 = 5 5 8.3

15. Now You Try P. 663, #23: Find the equation that satisfies these conditions: Foci (±3, 0), transverse axis length 4 8.3

16. Example Find the coordinates of the center, foci, and vertices, and the equations of the asymptotes of the graph of 4x 2 – y 2 + 24x + 4y + 28 = 0 4x 2 + 24x – y 2 + 4y = -28 4(x 2 + 6x ) - (y 2 -4y ) = -28 4(x+3) 2 – (y-2) 2 = 4 + 9 + 36 + 4 - 4 8.3

17. Example 4(x+3) 2 – (y-2) 2 = 4 a = 1 b = 2 c 2 = a 2 + b 2 c 2 = 1 + 4 c 2 = 5 Now let’s find the vertices, foci, and asymptotes on the graph. 8.3

18. Example a=1 b=2 (-3, 2) (-2, 2) (-4, 2) 8.3

19. Now You Try P. 664, #49: Find the center, vertices, and foci of 9x 2 – 4y 2 – 36x + 8y – 4 =0 8.3

20. Eccentricity Hyperbolas have eccentricities too. Since c > a, E > 1 8.3

21. Example Write the equation of the hyperbola with center at (-2, -4), a focus of (2,-4) and eccentricity c = distance from center to focus = 4 Since c= 4, a = 3 c 2 = a 2 + b 2 4 2 = 3 2 + b 2 b 2 = 7 9 7 8.3

22. Now You Try p. 663, #37: Find an equation in standard form for the hyperbola that satisfies the conditions: Center(-3,6), a=5, e=2, vertical transverse axis 8.3

23. Home Work P. 663-665, #2, 6, 24, 28, 32, 38, 44, 50, #63-68 8.3