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1. Conic Sections Hyperbolas
2. Definition The conic section formed by a plane which intersects both of the right conical surfaces
Formed whenor when the plane isparallel to the axis of the cone
3. Definition A hyperbola is the set of all points in the plane where
The difference between the distances
From two fixed points (foci)
Is a constant
4. Experimenting with Definition Turn on Explore geometric definition. A purple point will appear on the hyperbola, along with two line segments labeled L1 and L2. Drag the purple point around the hyperbola.
Do the lengths of L1 and L2 change?
What do you notice about the absolute value of the differences of the lengths?
How do these observations relate to the geometric definition of a hyperbola.
Observe the values of L1, L2, and the difference | L1 - L2 | as you vary the values of a and b.
How is the difference | L1 - L2 | related to the values of a and/or b? (Hint: Think about multiples.)
Determine the difference | L1 - L2 | for a hyperbola where a = 3 and b = 4. Use the Gizmo to check your answer.
5. Elements of An Ellipse Transverse axis
Line joining the intercepts
Passes throughcenter, perpendicularto transverse axis
Points where hyperbola intersects transverse axis
6. Elements of An Ellipse Transverse Axis
Length = 2a
Location (-c, 0), (c, 0)
Experiment with Pythagorean relationship
7. Equations of An Ellipse Given equations of ellipse
Centered at origin
Opening right and left
Equations of asymptotes
Opening up and down
Equations of asymptotes
8. Try It Out Find
9. More Trials and Tribulations Find the equation in standard form of the hyperbola that satisfies the stated conditions.
Vertices at (0, 2) and (0, -2), foci (0, 3) and (0, -3)
Foci (1, -2) and (7, -2)slope of an asymptote = 5/4
10. Assignment Hyperbola A
Exercise set 6.3
Exercises 1 ? 25 oddand 33 ? 45 odd
11. Conic Sections
12. Eccentricity of a Hyperbola The hyperbola can be wide or narrow
13. Eccentricity of a Hyperbola As with eccentricity of an hyperbola, the formula is
Note that for hyperbolas c > a
Thus eccentricity > 1
14. Try It Out If the vertices are (1, 6) and (1, 8) and the eccentricity is 5/2
Find the equation (standard form) of the hyperbola
The center of the hyperbola is at (-3, -3) and the conjugate axis has length 6, and the eccentricity = 2
Find two possible hyperbola equations
15. Application ? Locating Position For any point on a hyperbolic curve
Difference between distances to foci is constant.
Result: hyperbolas can be used to locate enemy guns
16. Application ? Locating Position The loran system navigator
equipped with a map that gives curves, called loran lines of position.
Navigators find the time interval between these curves,
Narrow down the area that their craft's position is in.
Then switch to a different pair of loran transmitters
Repeat the procedure
Find another curve representing the craft's position.
17. Construction Consider a the blue string
Keep markeragainst rulerand with stringtight
Keep end ofruler on focusF1 , string tied to other end
18. Graphing a Hyperbola on the TI As with the ellipse, the hyperbola is not a function
Possible to solve for y
Get two expressions
What happensif it opensright and left?
19. Graphing a Hyperbola on the TI Top and bottom of hyperbola branches are graphed separately
As with ellipsesyou must
20. Assignment Hyperbolas 2
Exercise Set 6.3
Exercises 27 ? 31 oddand 49 ? 63 odd