1. Conic Sections: Hyperbolas
Pre–Calculus PreAP/Dual, Revised ©2015
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11.2: Hyperbolas

2. Hyperbola from a Cone
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11.2: Hyperbolas

3. Real-Life Examples
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11.2: Hyperbolas

4. 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.
Latus Rectum: A line segment the focus and parallel to the directrix.
Eccentricity: Ratio to describe the shape of the conic e > 1
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11.2: Hyperbolas

5. 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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11.2: Hyperbolas

6. 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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11.2: Hyperbolas

7. 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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11.2: Hyperbolas

8. Horizontal Hyperbola
(0, b)
(–a, 0)
(a, 0) F
(–c, 0) F
(c, 0)
(h, k)
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
(0, –b)
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11.2: Hyperbolas

9. Vertical Hyperbola F
(c, 0)
(0, –a)
(–b, 0)
(b, 0)
(h, k)
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
(0, a) F
(–c, 0)
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11.2: Hyperbolas

10. Ellipses vs. hyperbolas
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11.2: Hyperbolas

11. When in Trouble…
PLOT and GRAPH
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11.2: Hyperbolas

12. Remember…
SIZE DOESN’T MATTER in hyperbolas. SIZE DOES MATTER in Ellipses. WHATEVER COMES FIRST is where the axis lies.
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11.2: Hyperbolas

13. Example 1 Graph 𝒙 𝟐 𝟐𝟓 − 𝒚 𝟐 𝟒 =𝟏 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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11.2: Hyperbolas

14. Example 2 Graph 𝒚 𝟐 𝟒 − 𝒙 𝟐 𝟐𝟓 =𝟏 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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11.2: Hyperbolas

15. 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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11.2: Hyperbolas

16. 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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11.2: Hyperbolas

17. Example 4
Write an equation where vertices are at (0, +6), asymptote is y= –6/7x and graph.
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11.2: Hyperbolas

18. 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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19. Example 5 Graph 𝒙−𝟑 𝟐 𝟗 − 𝒚−𝟐 𝟐 𝟒 =𝟏 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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11.2: Hyperbolas

20. Example 6 Graph 𝒚−𝟑 𝟐 𝟒 − 𝒙+𝟐 𝟐 𝟔 =𝟏 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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11.2: Hyperbolas

21. Your Turn Graph 𝒚−𝟑 𝟐 𝟒𝟗 − 𝒙+𝟏 𝟐 𝟐𝟓 =𝟏 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
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11.2: Hyperbolas

22. 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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11.2: Hyperbolas

23. 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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11.2: Hyperbolas

24. Assignment
Worksheet
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11.2: Hyperbolas