1. WORKSHEET KEY
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9.5: Hyperbolas

2. WORKSHEET KEY
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9.5: Hyperbolas

4. Hyperbolas: Conic Section
Revised ©2013,
Hyperbolas: Conic Section
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9.5: Hyperbolas

5. Hyperbola from a Cone
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9.5: Hyperbolas

6. Real-Life Examples
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7. Definitions
Hyperbola: Set of points whose differences of the distances from any point to the foci is constant. It has two disconnected branches.
Foci: Segment point joining the vertices given at a point. It is always with the TRANSVERSE axis
Transverse Axis: The line segment joining the vertices.
Conjugate Axis: The minor line segment joining the vertices perpendicular to the transverse axis.
Asymptote: Line that a graph approaches but does not ever intersect. If the lines intersect, it becomes undefined.
Eccentricity: Ratio to describe the shape of the conic e > 1
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8. FORMULAS TO KNOW If the transverse axis is on the x-axis (horizontal)
Horizontal Axis Standard Form:
If the transverse axis is on the x-axis (horizontal)
If the transverse axis is on the x-axis (horizontal)
a is associated with transverse axis because it comes first.
Foci of the transverse axis:
Asymptotes:
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9. FORMULAS TO KNOW If the transverse axis is on the y-axis (vertical)
Vertical Axis Standard Form:
If the transverse axis is on the y-axis (vertical)
If the transverse axis is on the y-axis (vertical)
a is associated with transverse axis because it comes first.
Foci of the transverse axis:
Asymptotes:
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9.5: Hyperbolas

10. All Standard Form Equations
FORMULAS TO KNOW
All Standard Form Equations
Center
Length of Transverse Axis
Length of Conjugate Axis
Foci Equation
Length of Latus Rectum
Eccentricity
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11. Horizontal Hyperbola
(0, b)
(–a, 0)
(a, 0) F
(–c, 0) F
(c, 0)
(h, k)
(0, –b)
Center: (h, k) Foci: (+c, 0) Length of Transverse Axis: 2a Vertices: (+a, 0) Length of Conjugate Axis: 2b Co-Vertices: (0, +b) Asymptotes: y – k = +b/a(x – h) Length of Latus Rectum: 2b2/a Latus Rectum: (+c, +b2/a) Eccentricity: c/a
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12. Vertical Hyperbola F
(c, 0)
(0, –a)
(–b, 0)
(b, 0)
(h, k)
(0, a)
Center: (h, k) Foci: (0, +c) Length of Transverse Axis: 2a Vertices: (0, +a) Length of Conjugate Axis: 2b Co-Vertices: (+b, 0) Asymptotes: y – k = +a/b(x – h) Length of Latus Rectum: 2b2/a Latus Rectum: (+b2/a, +c) Eccentricity: c/a F
(–c, 0)
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9.5: Hyperbolas

13. Ellipses vs. hyperbolas
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14. When in Trouble…
PLOT and GRAPH
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15. Remember… SIZE DOESN’T MATTER in hyperbolas.
SIZE DOES MATTER in Ellipses.
WHATEVER COMES FIRST is where the axis lies.
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16. Example 1 Graph F F Type: Center: Vertices: Co-Vertices: Asymptotes:B: C:
Type:
Center:
Vertices:
Co-Vertices:
Asymptotes:
Foci:
Latus Rectum:
Length of Transverse Axis:
Length of Conjugate Axis:
Length of Latus Rectum:
Eccentricity: F F
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17. Example 2 Graph F F Type: Center: Vertices: Co-Vertices: Asymptotes:B: C:
Type:
Center:
Vertices:
Co-Vertices:
Asymptotes:
Foci:
Latus Rectum:
Length of Transverse Axis:
Length of Conjugate Axis:
Length of Latus Rectum:
Eccentricity: F F
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18. YOUR TURN Graph 49y2 – 36x2 = 1764 F F Type: Center: Vertices:B: C:
Type:
Center:
Vertices:
Co-Vertices:
Asymptotes:
Foci:
Latus Rectum:
Length of Transverse Axis:
Length of Conjugate Axis:
Length of Latus Rectum:
Eccentricity: F F
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19. Example 3
Write an equation of a hyperbola where the foci is (+3, 0) and the vertices are (+2, 0). F F
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20. Example 4
Write an equation where vertices are at (0, +6), asymptote is y= –6/7x and graph.
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21. YOUR TURN
Write an equation where center is (0, 0) opens vertically. Points used to make the rectangle are (0, +4) and (+6, 0).
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22. Example 5 Graph F F Type: Center: Vertices: Co-Vertices: Asymptotes:B: C:
Type:
Center:
Vertices:
Co-Vertices:
Asymptotes:
Foci:
Latus Rectum:
Length of Transverse Axis:
Length of Conjugate Axis:
Length of Latus Rectum:
Eccentricity: F F
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23. Example 6 Graph F F Type: Center: Vertices: Co-Vertices: Asymptotes:B: C:
Type:
Center:
Vertices:
Co-Vertices:
Asymptotes:
Foci:
Latus Rectum:
Length of Transverse Axis:
Length of Conjugate Axis:
Length of Latus Rectum:
Eccentricity: F F
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24. YOUR TURN Graph F F Type: Center: Vertices: Co-Vertices: Asymptotes:B: C:
Type:
Center:
Vertices:
Co-Vertices:
Asymptotes:
Foci:
Latus Rectum:
Length of Transverse Axis:
Length of Conjugate Axis:
Length of Latus Rectum:
Eccentricity: F F
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25. Example 7
Write an equation where the foci coordinates are (6, 2) & (–8, 2) and vertices are (2, 2) & (–4, 2). F F
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26. Example 8
Write an equation where center is (–3, 7), foci coordinates are (–3 + √85, 7) and vertices are (–5, 7) & (–1, 7).
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27. YOUR TURN
Write an equation where center is (1, –2), foci coordinates are (1 + √5, –2) and vertices are (3, –2) & (–1, –2).
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28. CONSTANT DIFFERENCE Apply into the equation d = |PF1 – PF2|
Determine the distance using the Distance Formula of the points,
Simplify
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29. Example 9
Find the constant difference for a hyperbola with foci F1 (–8, 0) and F2 (8, 0) and the point on the hyperbola (8, 30).
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30. Example 9
Find the constant difference for a hyperbola with foci F1 (–8, 0) and F2 (8, 0) and the point on the hyperbola (8, 30). 34 30 F1 F2
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31. Your Turn
Find the constant difference for a hyperbola with foci F1 (0, –10) and F2 (0, 10) and the point on the hyperbola (6, 7.5).
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9.5: Hyperbolas

32. ASSIGNMENT
Worksheet
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9.5: Hyperbolas