1. Conic Sections

2. Four conic sections: A plane intersecting a double cone will generate:
Circles
Parabolas
Ellipses
Hyperbolas
Parabola
Circle
Ellipse
Hyperbola

3. Circle

4. Circle
The Standard Form of a circle with a center at (h,k) and a radius, r
                                                                                                                                                                          center (3,3) radius = 2

5. Parabola

6. Parabola
A parabola is the set of all points in the plane that are equidistant from a fixed line (directrix) and a fixed point (focus) not on the line.
Demo

7. Why is the focus so important?

8. Parabola with vertical axis of symmetry
Standard equation: (x – h)2 = 4p(y – k)
p ≠ 0
the parabola opens: UP if p > 0, DOWN if p < 0 Vertex: (h, k) Focus: (h, k+p) Directrix: y = k - p Axis of symmetry: x = h

9. Parabola with horizontal axis of symmetry
Standard equation:(y – k)2 = 4p(x – h)
p ≠ 0
the parabola opens: RIGHT if p > 0, LEFT if p < 0 Vertex: (h, k) Focus: (h + p, k) Directrix: x = h – p Axis of symmetry: y = k

10. Practice – Parabola Find the focus, vertex and directrix:
3x + 2y2 + 8y – 4 = 0
Find the equation in standard form of a parabola with directrix x = -1 and focus (3, 2).
Find the equation in standard form of a parabola with vertex at the origin and focus (5, 0).
Work as a group

11. Ellipse © Jill Britton, September 25, 2003
Statuary Hall in the U.S. Capital building is elliptic. It was in this room that John Quincy Adams, while a member of the House of Representatives, discovered this acoustical phenomenon. He situated his desk at a focal point of the elliptical ceiling, easily eavesdropping on the private conversations of other House members located near the other focal point.

12. Ellipse
The ellipse is the set of all points in the plane for which the sum of distances from two fixed points (foci) is a positive constant.

13. Parts of an Ellipse: Major axis - longer axis, contains foci
Minor axis - shorter axis
Semi-axis - ½ the length of axis
Center - midpoint of major axis
Vertices - endpoints of the major axis
Foci - two given points on the major axis
Emphasis the relationship between a, b, and c…..a > c
Focus
Center
Focus

14. Why are the foci of the ellipse important?
St. Paul's Cathedral in London. If a person whispers near one focus, he can be heard at the other focus, although he cannot be heard at many places in between.

15. Why are the foci of the ellipse important?
The ellipse has an important property that is used in the reflection of light and sound waves. Any light or signal that starts at one focus will be reflected to the other focus. This principle is used in lithotripsy, a medical procedure for treating kidney stones. The patient is placed in a elliptical tank of water, with the kidney stone at one focus. High-energy shock waves generated at the other focus are concentrated on the stone, pulverizing it.

16. Ellipse with horizontal major axis:
Center is (h, k). Length of major axis is 2a. Length of minor axis is 2b. Distance between center and either focus is c with  c2 = a2– b2, a > b > 0.

17. Ellipse with vertical major axis:
Center is (h, k). Length of major axis is 2a. Length of minor axis is 2b. Distance between center and either focus is c with  c2 = a2– b2, a > b > 0.

18. Practice – Ellipse: Graph 4x 2 + 9y2 = 4
Find the vertices and foci of an ellipse - sketch the graph:
4x2 + 9y2 – 8x + 36y + 4 = 0
put in standard form
find center, vertices, and foci
Write the equation of the ellipse with:
center at (4, -2), the foci are (4, 1) and (4, -5) and the length of the minor axis is 10.
Work as a group

19. Hyperbola
The huge chimney of a nuclear power plant has the shape of a hyperboloid, as does
the architecture of the James S. McDonnell Planetarium of the St. Louis Science Center.

20. Where are the Hyperbolas?
A sonic boom shock wave has the shape of a cone, and it intersects the ground in part of a hyperbola. It hits every point on this curve at the same time, so that people in different places along the curve on the ground hear it at the same time. Because the airplane is moving forward, the hyperbolic curve moves forward and eventually the boom can be heard by everyone in its path.

21. Hyperbola
The hyperbola is the set of all points in a plane for which the difference between the distances from two fixed points (foci) is a positive constant.
Emphasis on difference vs sum for ellipses
Differs from an Ellipse whose sum of the distances was a constant.

22. Parts of the hyperbola:
Transverse axis
Conjugate axis
Vertices
Foci (will be on the transverse axis)
Center
Asymptotes

23. Hyperbola with horizontal transverse axis
Center is (h, k). Distance between the vertices is 2a. Distance between the foci is 2c. c2 = a2 + b2

24. Eccentricity of the hyperbola:
Eccentricity e = c/a since c > a , e >1
As the eccentricity gets larger the graph becomes wider and wider

25. Hyperbola with vertical transverse axis- +
Center is (h, k). Distance between the vertices is 2b. Distance between the foci is 2c. c2 = a2 + b2

26. Practice – Hyperbola: Write in standard form: 9y2 – 25x2 = 225
4x2 –25y2 +16x +50y –109 = 0
Emphasize relationship between a, b, c…… c > a
Work as a group

27. Conclusion:
Most General Equation of a Conic Section:

28. Which Conic is it? Parabola: A = 0 OR C = 0 Circle: A = C
Ellipse: A ≠ C, but both have the same sign
Hyperbola: A and C have Different signs

29. Example: State the type of conic and write it in the standard form of that conic
Conic: A and C same sign, but A ≠ C
ELLIPSE
Standard Form:

30. Example: State the type of conic and write it in the standard form of that conic
PARABOLA
Standard Form:
(y – k)2 = 4p(x – h)

31. Example: State the type of conic and write it in the standard form of that conic
Conic: A =C
CIRCLE
Standard Form:
**Divide all by 2!**

32. HW assignment - Identify, write in standard form and graph each of the following:
4x2 + 9y2-16x - 36y -16 = 0
2x2 +3y - 8x + 2 =0
5x - 4y =0
9x2 + 25y2 - 18x +50y = 0
2x2 + 2y2 = 10
(x+1)2 + (y- 4) 2 = (x + 3)2
Work as a group

34. Additional images