11111Q1 - INTRODUCTION TO CONIC SECTIONS.pptx

PRE CALCULUS

1. PARABOLA

2. CIRCLE

3. PARABOLA

4. HYPERBOLA

5. ELLIPSE

6. HYPERBOLA

7. Precalculus Science, Technology, Engineering, and Mathematics Lesson 1.1 Introduction to Conic Sections

8. The planets, comets, and stars take a path that corresponds to one of the conic sections, which are: ellipse, parabola, hyperbola, and circle. 8

9. 9

10. 10 What does each conic section look like?

11. 11 Illustrate the different types of conic sections: parabola, ellipse, circle, hyperbola, and degenerate cases (STEM_PC11AG-ia-1)

12. 12 Generate conic sections from the intersection of a plane and a cone. Identify the conic sections: parabola, ellipse, circle, hyperbola, and degenerate cases. Locate the common parts of the conic sections.

13. 13 Conic sections are obtained from the intersection between a double-napped cone (right circular cone) and a plane. Conic Sections

14. 14 Conic Sections Slanting Height Altitude

15. 15 Circles are formed when the intersection of the plane is perpendicular to the axis of revolution. Circle

16. 16 Ellipses are formed when the (tilted) plane intersects only one cone to form a bounded curve. Ellipse

17. 17 Parabolas are formed when the plane is parallel to the generating line of one cone. Parabola

18. 18 Hyperbolas are formed when the plane is parallel to the axis of revolution or the -axis   Hyperbola

19. 19 Degenerate conic sections are formed when a plane intersects the cone in such a way that it passes through the apex (vertex). Degenerate Conic Sections

20. 20 Degenerate Conic Sections

21. 21 Degenerate Conic Sections Single Point

22. 22 Degenerate Conic Sections Single Line

23. 23 Degenerate Conic Sections Two Intersecting lines

24. 24 Degenerate Conic Sections

25. 25 Common Parts of the Conic Sections Vertex (with horizontal axis) - an extreme point on a parabola, hyperbola, and ellipse

26. 26 Common Parts of the Conic Sections Vertex (with vertical axis) - an extreme point on a parabola, hyperbola, and ellipse

27. 27 Common Parts of the Conic Sections Focus and Directrix (with horizontal axis) These are the point and the line on a conic section that are used to define and construct the curve, respectively.

28. 28 Common Parts of the Conic Sections Focus and Directrix (with vertical axis) These are the point and the line on a conic section that are used to define and construct the curve, respectively.

29. 29 Common Parts of the Conic Sections Center It is the midpoint between the two foci of an ellipse and hyperbola.

30. 30 Common Parts of the Conic Sections Center For circles , center is the point equidistant from any point on the surface.

31. 31 Given the curve on the Cartesian plane, identify the vertex, focus, and directrix.

32. 32 Given the curve on the Cartesian plane, identify the vertex, focus, and directrix. Vertex : ; Focus : ; Directrix :  

33. 33 Identify the coordinates of the foci and center of the graph below.

34. 34 Identify the coordinates of the foci and center of the graph below. Foci : ; Center :  

35. 35 What are the different conic sections and their common parts?

36. 36 If a cone shaped pita bread was cut as shown in the figure on the right, which curve will be formed between the intersection of the knife and the pita bread?

37. 37 If a cone shaped pita bread was cut as shown in the figure below, which curve will be formed between the intersection of the knife and the pita bread? parabola

38. 38 38 An ice cream cone was cut by a knife to get only the bottom part filled with chocolates as shown below. What curve was formed between the intersection of the knife and the ice cream cone?  

39. 39 Plot the curve of the Gateway Arch in St. Louis Missouri, United States on a Cartesian plane if its vertex is at the origin, with a focus at .  

40. 40 Plot the curve of the Gateway Arch in St. Louis Missouri, United States on a Cartesian plane if its vertex is at the origin, with a focus at .  

41. 41 Plot the curve of the Gateway Arch in St. Louis Missouri, United States on a Cartesian plane if its vertex is at the origin, with a focus at . Give the type of conic and solve for its directrix.   The d irectrix is  

42. 42 42 Plot this plane figure of a football on a Cartesian Plane. If the length of the football is 12 in, height is 8 in, center at , and foci at and , graph and give the type of conic.  

43. 43 Identify the conic section or the part that is being described. 1. The se are the conic sections that are formed when the plane intersects the double-napped cone in a way that it passes through the apex. 2. This conic section is formed when the plane is parallel to the axis of revolution. 3. It is the midpoint of the two foci for ellipse and hyperbola.

44. 44 Using the image, complete the table and solve for the directrix given the vertices and foci.

45. 45 Conics Vertex/ Vertices Focus/ Foci Directrix (1) (4) (2) (5) (3) (6) Conics Vertex/ Vertices Focus/ Foci Directrix (1) (4) (2) (5) (3) (6)

46. 46 Analyze and solve the problem below. Make an approximate sketch of the curve of the Eiffel Tower on the cartesian plane, with its center at , and say that the vertices is at and the foci is at Give the type of conic section, and its directrix.  

47. 47 Conic sections are curves obtained from the intersection between a double-napped cone and a plane. There are basically three types of conic sections: parabola , hyperbola , and ellipse . A circle is a type of ellipse and is sometimes considered as the fourth conic section.

48. 48 A parabola is formed when the plane is parallel to the generating line of one cone. An ellipse is formed when the plane intersects the cone at an angle other than . A hyperbola is formed when the plane is parallel to the axis of revolution or the 𝑦-axis.  

49. 49 A circle is formed when the intersection of the plane is perpendicular to the axis of revolution. Degenerate conic sections are formed when the plane intersects the cone in such a way that it passes through the apex.

50. 50 The conic sections have common parts, which are the vertex , the focus , directrix , and the center for ellipse and hyperbola. Vertex is an extreme point on a parabola and hyperbola.

51. 51 The focus and directrix are the point and the line on a conic section that are used to define and construct the curve, respectively. Center is the midpoint between the two foci of an ellipse and hyperbola. For circles, the center is the point equidistant from any point on the surface.

52. 52 52 A glass was placed on the table. If you hold a flashlight as shown below, what kind of curve will be formed by its shadow?

53. 53 Slide no.2: 01 The Solar System PIA10231, mod02 by Image Editor is licensed under CC By 2.0 via Flickr . Slide no.32: Gateway Arch St. Louis from Illinois by Mobilus In Mobili is licensed under CC BY-SA 2.0 via Flickr . Slide no.35: American Football 1.svg by feraliminal is licensed under CC0 1.0 via Wikimedia Commons.

54. 54 Boeckmann , Catherine. “What Are Perihelion and Aphelion?” Old Farmer's Almanac. Accessed January 7, 2020 from https://www.almanac.com/content/what-aphelion-and-perihelion .   “Conic Section Directrix.” Wolfram MathWorld . Accessed December 6, 2019 from http://mathworld.wolfram.com/ConicSectionDirectrix.html#:~:targetText=The%20directrix%20of%20a%20conic,being%20the%20constant%20of%20proportionality .   “Introduction to Conic Sections.” Lumen. Accessed December 5, 2019 from https://courses.lumenlearning.com/boundless-algebra/chapter/introduction-to-conic-sections/ .   James Stewart, Lothar Redlin, and Saleem Watson, Precalculus Mathematics for Calculus , 7 th Edition (Boston, MA: Cengage Learning, 2016).   Ron Larson, Precalculus, 9 th Edition (Boston, MA: Cengage Learning, 2013). The Editors of Encyclopaedia Britannica. “Kepler's Laws of Planetary Motion.” Encyclopædia Britannica. Encyclopædia Britannica, inc. , October 31, 2019. https://www.britannica.com/science/Keplers-laws-of-planetary-motion .