Introduction to Conic Sections

1. Precalculus / Basic Calculus
Science, Technology, Engineering, and Mathematics
Precalculus
Science, Technology, Engineering, and Mathematics
Lesson 1.1
Introduction to Conic
Sections

2. The planets,
comets, and stars
take a path that
corresponds to
one of the conic
sections, which
are: ellipse,
parabola,
hyperbola, and
circle.
2

3. 3
What does each conic section
look like?

4. Learning Competency
At the end of the lesson, you should be able to do the following:
4
Illustrate the different types of conic sections:
parabola, ellipse, circle, hyperbola, and
degenerate cases (STEM_PC11AG-ia-1)

5. Learning Objectives
At the end of the lesson, you should be able to do the following:
5
● 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.

6. 6
Conic sections are obtained from the intersection
between a double-napped cone and a plane.
Conic Sections

7. 7
Parabolas are formed when the plane is parallel to the
generating line of one cone.
Parabola

8. 8
Ellipses are formed when the plane intersects the one
cone at an angle other than 90°.
Ellipse

9. 9
Hyperbolas are formed when the plane is parallel to the
axis of revolution or the 𝑦-axis.
Hyperbola

10. 10
Circles are formed when the intersection of the plane is
perpendicular to the axis of revolution.
Circle

11. 11
Degenerate conic sections are formed when a plane
intersects the cone in such a way that it passes through
the apex.
Degenerate Conic Sections

12. 12
Degenerate Conic Sections
Two Intersecting lines

13. 13
Degenerate Conic Sections
Single Line

14. 14
Degenerate Conic Sections
Single Point

15. 15
Degenerate Conic Sections

16. 16
Common Parts of the Conic Sections
Vertex (with horizontal axis)
- an extreme point on a parabola, hyperbola, and ellipse

17. 17
Common Parts of the Conic Sections
Vertex (with vertical axis)
- an extreme point on a parabola, hyperbola, and ellipse

18. 18
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.

19. 19
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.

20. 20
Common Parts of the Conic Sections
Center
It is the midpoint between the two foci of an ellipse and
hyperbola.

21. 21
Common Parts of the Conic Sections
Center
For circles, center is the point
equidistant from any point on
the surface.

22. 22
What are the different conic
sections and their common
parts?

23. Let’s Practice!
23
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?

24. Let’s Practice!
24
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

25. Try It!
25
25
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?

26. Let’s Practice!
26
Given the curve on the
Cartesian plane, identify
the vertex, focus, and
directrix.

27. Let’s Practice!
27
Given the curve on the Cartesian plane, identify the
vertex, focus, and directrix.
Vertex : (𝟎, 𝟎) ; Focus : (𝟑, 𝟎) ; Directrix : 𝒙 = −𝟑

28. Try It!
28
28
Given the curve on the Cartesian plane,
identify the focus, vertex, and directrix.

29. Let’s Practice!
29
Identify the coordinates of the foci and center of
the graph below.

30. Let’s Practice!
30
Identify the coordinates of the foci and center of the
graph below.
Foci : (−𝟐, −𝟑) , (𝟔, −𝟑) ; Center : (𝟐, −𝟑)

31. Try It!
31
31
Identify the foci and the center of the
graph below.

32. Let’s Practice!
32
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 (𝟎, −𝟑). Give the type of conic and solve for its directrix.

33. Let’s Practice!
33
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 (𝟎, −𝟑). Give the type of conic and solve for its directrix.

34. Let’s Practice!
34
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 (𝟎, −𝟑). Give the
type of conic and solve for its directrix.
The directrix is 𝒚 = 𝟑.

35. Try It!
35
35
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
(𝟎, 𝟎), and foci at (−𝟐 𝟓, 𝟎) and (𝟐 𝟓, 𝟎),
give the type of conic, and solve for its
directrix.

36. Check Your Understanding
36
Identify the conic section or the part that is being
described.
1. These 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.

37. Check Your Understanding
37
Using the image,
complete the table and
solve for the directrix
given the vertices and
foci.

38. Check Your Understanding
38
Conics
Vertex/
Vertices
Focus/
Foci
Directrix
(1) (0,0) −
5
4
, 0 (4)
(2)
−5, 0
5, 0
− 61, 0
61, 0
(5)
(3)
−3,0
(3,0)
− 5, 0
5, 0
(6)

39. Check Your Understanding
39
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 (0,0), and
say that the vertices is at −2, 0 , (2, 0), and the foci is at
−2 17, 0 , 2 17, 0 . Give the type of conic section, and its
directrix.

40. Let’s Sum It Up!
40
● 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.

41. Let’s Sum It Up!
41
● 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 90°.
● A hyperbola is formed when the plane is parallel
to the axis of revolution or the 𝑦-axis.

42. Let’s Sum It Up!
42
● 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.

43. Let’s Sum It Up!
43
● 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.

44. Let’s Sum It Up!
44
● 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.

45. Challenge Yourself
45
45
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?

46. Photo Credits
46
● 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.

47. Bibliography
47
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%20
a%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, 7th Edition
(Boston, MA: Cengage Learning, 2016).
Ron Larson, Precalculus, 9th 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.