This document provides information about hyperbolas and their applications. It discusses key concepts such as the definition and properties of hyperbolas, including their standard equation forms. Real-world applications like navigation systems and cooling tower design are explained. Examples of word problems involving hyperbolas are worked through, covering finding equations and distances. Formulas for relationships between hyperbola parameters are also reviewed.

1. Capstone Project
Science, Technology, Engineering, and Mathematics
Precalculus
Science, Technology, Engineering, and Mathematics
Lesson 4.3
Applications of
Hyperbolas in Real-life
Situations

2. 2
Have you every
wondered how we
were able to track the
locations of ships and
aircrafts before we
had navigation
satellites (e.g. Global
Positioning System or
GPS)?

3. 3
Navigation systems
such as the LORAN
(long-range
navigation) use radio
signals to determine
the exact location of a
ship or aircraft.

4. 4
Knowledge of hyperbolas is used in this type of navigation
system.

5. 5
What are some real-life
applications of hyperbolas?

6. Learning Competency
At the end of the lesson, you should be able to do the following:
6
Solve situational problems involving
hyperbolas (STEM_PC11AG-Ie-2).

7. Learning Objectives
At the end of the lesson, you should be able to do the following:
7
● Recall the different parts and properties of a
hyperbola.
● Solve word problems on hyperbolas.

8. 8
A hyperbola is defined as a set of points on a plane
whose absolute difference between the distances from
the foci, 𝐹1 and 𝐹2, is constant.
Hyperbolas

9. 9
Parts and Properties of Hyperbolas
Given arbitrary points 𝑃1 and 𝑃2 on the hyperbola,
|𝑃1𝐹1 − 𝑃1𝐹2| = 𝑃2𝐹1 − 𝑃2𝐹2 = 2𝑎.
What is the distance 𝑎?

10. 10
Parts and Properties of Hyperbolas
𝒄2 = 𝒂2 + 𝒃2

11. 11
How do you determine the value
of 𝒂 given the values of 𝒃 and 𝒄?
How do you determine the value
of 𝒃 given the values of 𝒂 and 𝒄?

12. 12
Standard Equations of a Hyperbola
Orientation of
Principal Axis
Equation
Horizontal
𝒙 − 𝒉 𝟐
𝒂𝟐
−
𝒚 − 𝒌 𝟐
𝒃𝟐
= 𝟏
Vertical
𝒚 − 𝒌 𝟐
𝒂𝟐
−
𝒙 − 𝒉 𝟐
𝒃𝟐
= 𝟏

13. 13
Applications of Hyperbolas
Hyperbolic
Navigation
Example: LORAN-C
(long-range navigation)

14. 14
Applications of Hyperbolas
Hyperbolic
Navigation
Two stations, 𝐴 and 𝐵,
transmit radio signals.
The radio signals reach
the ship at different
times.

15. 15
Applications of Hyperbolas
Hyperbolic
Navigation
The location 𝑃 of the
ship is somewhere on
a hyperbola.

16. 16
Applications of Hyperbolas
Hyperbolic
Navigation
The difference
between the distances
of the two stations
from the ship at point
𝑃 is 𝑷𝑨 − 𝑷𝑩 = 𝟐𝒂.

17. 17
Applications of Hyperbolas
Cooling Towers
Cooling towers use
water from rivers and
lakes to cool nuclear
power plants.

18. 18
Applications of Hyperbolas
Cooling Towers
They protect aquatic
life by ensuring that
the water is returned
to the environment at
normal temperatures.

19. 19
Applications of Hyperbolas
Cooling Towers
Cooling towers are
hyperboloid in shape.
Why?

20. 20
Applications of Hyperbolas
Cooling Towers
 able to withstand
strong winds
 efficient in cooling
 economical

21. Let’s Practice!
21
Point 𝑷 is on a hyperbola. The distance of 𝑷 from the
first focus is 6 units more than its distance from the
second focus. What is the distance of the center to a
vertex of the hyperbola?

22. Let’s Practice!
22
3 units
Point 𝑷 is on a hyperbola. The distance of 𝑷 from the
first focus is 6 units more than its distance from the
second focus. What is the distance of the center to a
vertex of the hyperbola?

23. Try It!
23
23
Point 𝑷 is on a hyperbola. The
difference between the distances of 𝑷
from the first focus and the second
focus is 20 units. What is the distance
of the center to a vertex of the
hyperbola?

24. Let’s Practice!
24
What is the equation of a hyperbola whose center is
at the origin, has a horizontal principal axis, the
value of 𝒂 is 𝟒, and the point (𝟗, 𝟏𝟒) is on the
hyperbola?

25. Let’s Practice!
25
𝒙𝟐
𝟏𝟔
−
𝒚𝟐
𝟒𝟖. 𝟐𝟓
= 𝟏
What is the equation of a hyperbola whose center is
at the origin, has a horizontal principal axis, the
value of 𝒂 is 𝟒, and the point (𝟗, 𝟏𝟒) is on the
hyperbola?

26. Try It!
26
26
What is the equation of a hyperbola
whose center is at the origin, has a
vertical principal axis, the value of 𝒂
is 𝟖, and the point (𝟖, 𝟏𝟑) is on the
hyperbola?

27. Let’s Practice!
27
Point 𝑷 is on a hyperbola with a vertical principal
axis and whose center is at the origin. The distance
of 𝑷 from the first focus 𝑭𝟏 is 12 units more than its
distance from the second focus 𝑭𝟐. The distance
between the two foci is 20 units. What is the
equation of the hyperbola?

28. Let’s Practice!
28
𝒚𝟐
𝟑𝟔
−
𝒙𝟐
𝟔𝟒
= 𝟏
Point 𝑷 is on a hyperbola with a vertical principal
axis and whose center is at the origin. The distance
of 𝑷 from the first focus 𝑭𝟏 is 12 units more than its
distance from the second focus 𝑭𝟐. The distance
between the two foci is 20 units. What is the
equation of the hyperbola?

29. Try It!
29
29
Point 𝑷 is on a hyperbola with a
vertical principal axis and whose
center is at the origin. The distance of
𝑷 from the first focus 𝑭𝟏 is 10 units
more than its distance from the
second focus 𝑭𝟐. The distance between
the two foci is 26 units. What is the
equation of the hyperbola?

30. Let’s Practice!
30
The sides of a cooling tower represent a hyperbola. The
width of the slimmest part of the cooling tower is 50
meters. The length from this point to the top is 64
meters. The diameter of the top of the cooling tower is
60 meters. Find the equation of the hyperbola that
represents the sides of the cooling tower. Set the
middle of the slimmest part as the origin.

31. Let’s Practice!
31
𝒙𝟐
𝟔𝟐𝟓
−
𝒚𝟐
𝟗𝟑𝟎𝟗. 𝟎𝟗
= 𝟏
The sides of a cooling tower represent a hyperbola. The
width of the slimmest part of the cooling tower is 50
meters. The length from this point to the top is 64
meters. The diameter of the top of the cooling tower is
60 meters. Find the equation of the hyperbola that
represents the sides of the cooling tower. Set the
middle of the slimmest part as the origin.

32. Try It!
32
32
The sides of a cooling tower represent a
hyperbola. The width of the slimmest part
of the cooling tower is 54 meters. The
length from this point to the top is 70
meters. The diameter of the top of the
cooling tower is 68 meters. Find the
equation of the hyperbola that represents
the sides of the cooling tower. Set the
middle of the slimmest part as the origin.

33. Let’s Practice!
33
Two stations 𝑨 and 𝑩 are along a straight coast such
that station 𝑨 is 100 miles west of station 𝑩. They both
transmit radio signals at the speed of 186 000 miles per
second. A ship sailing at sea is 50 miles from the coast.
The radio signal from station 𝑩 arrives at the ship
0.0002 of a second earlier than the radio signal sent
from station 𝑨. Where is the ship? Set the midpoint of 𝑨
and 𝑩 as the origin.

34. Let’s Practice!
34
(𝟐𝟕. 𝟑𝟒, 𝟓𝟎)
Two stations 𝑨 and 𝑩 are along a straight coast such
that station 𝑨 is 100 miles west of station 𝑩. They both
transmit radio signals at the speed of 186 000 miles per
second. A ship sailing at sea is 50 miles from the coast.
The radio signal from station 𝑩 arrives at the ship
0.0002 of a second earlier than the radio signal sent
from station 𝑨. Where is the ship? Set the midpoint of 𝑨
and 𝑩 as the origin.

35. Try It!
35
35
Two stations 𝑨 and 𝑩 transmit radio signals
such that station A is 200 miles west of station
𝑩. Both stations sent radio signals with a speed
of 𝟎. 𝟐 miles per microsecond to a plane
traveling west. The signal from station 𝑩
reaches a plane 500 microseconds faster than
the signal from station 𝑨. If the plane is 80
miles north of the line from stations 𝑨 to 𝑩,
what is the location of the plane? Set the
midpoint of 𝑨 and 𝑩 as the origin.

36. Check Your Understanding
36
Let 𝑷 be a point on a hyperbola with center at the
origin and foci 𝑭𝟏 and 𝑭𝟐. Answer the following
questions. Round off your answer to two decimal
places.
1. If 𝑃𝐹1 − 𝑃𝐹2 = 18, what is 𝑎?
2. If 𝑃𝐹2 − 𝑃𝐹1 = 32 and 𝑏 = 12, what is 𝑐?
3. If 𝑃𝐹1 − 𝑃𝐹2 = 34 and 𝑏2 = 256, what is 𝑐?
4. If 𝑐 = 39 and 𝑎 = 36, what is 𝑏?
5. If 𝑐2 = 1 600 and 𝑎 = 32, what is 𝑏?

37. Check Your Understanding
37
Solve the following problems.
1. A nuclear power plant has a cooling tower whose sides
are in the shape of a hyperbola. The slimmest part of
the tower is 58 meters. The length from this point to
the top is 80 meters. The radius of the top of the
cooling tower is 72 meters. What is the equation of the
hyperbola that represents the sides of the cooling
tower? Set the middle of the slimmest part as the
origin.

38. Check Your Understanding
38
Solve the following problems.
2. The sides of a cooling tower represent a hyperbola. The
width of the slimmest part of the cooling tower is 80
meters. The slimmest part of the tower is 90 meters
from the ground. The diameter of the base of the tower
is 130 meters. What is the equation of the hyperbola
that represents the sides of the cooling tower? Set the
middle of the slimmest part as the origin.

39. Check Your Understanding
39
Solve the following problems.
3. Two radio signaling stations 𝐴 and 𝐵 are 120 kilometers
apart. The radio signal from the two stations has a
speed of 300 000 kilometers per second. A ship at sea
receives the signals such that the signal from station 𝐵
arrives 0.0002 seconds before the signal from station 𝐴.
What is the equation of the hyperbola where the ship is
located? Set the midpoint of 𝐴 and 𝐵 as the origin.

40. Let’s Sum It Up!
40
● A hyperbola is a set of points on a plane whose
absolute difference between the distances from two
fixed points 𝐹1 and 𝐹2 is constant.
● Given any point 𝑃 on a hyperbola with foci 𝐹1 and 𝐹2,
𝑃𝐹1 − 𝑃𝐹2 = 2𝑎.

41. Let’s Sum It Up!
41
● The distance from the center to a vertex of the
hyperbola is 𝒂, the distance from the center to an
endpoint of the conjugate axis is 𝒃, and the focal
distance is 𝒄.
● The relationship between the three distances is
𝒂𝟐 + 𝒃𝟐 = 𝒄𝟐.

42. Let’s Sum It Up!
42
● The standard forms of equation of a hyperbola whose
center is at the origin are
𝒙𝟐
𝒂𝟐 −
𝒚𝟐
𝒃𝟐 = 𝟏 if the principal axis
is horizontal, and
𝒚𝟐
𝒂𝟐 −
𝒙𝟐
𝒃𝟐 = 𝟏 if the principal axis is
vertical.

43. Let’s Sum It Up!
43
● In the real world, hyperbolas are used for navigation
and the structure of cooling towers.

44. Let’s Sum It Up!
44
● To solve word problems on hyperbolas, follow these
steps:
1. Determine the standard form of equation of the
hyperbola.
2. Draw an illustration to be able to visualize the
problem.
3. Solve for the unknown equation or values.

45. Key Formulas
45
Concept Formula Description
Relationship between
the values 𝒂, 𝒃, and 𝒄
𝑎2 + 𝑏2 = 𝑐2,
where 𝑎 is the distance
from the center to a
vertex, 𝑏 is the distance
from the center to an
endpoint of the
conjugate axis, and 𝑐 is
the focal distance.
Use this formula to find
an unknown distance
(e.g., focal distance)
when the other values
are known.

46. Key Formulas
46
Concept Formula Description
Equation of a
Hyperbola in Standard
Form
𝑥 − ℎ 2
𝑎2 −
𝑦 − 𝑘 2
𝑏2 = 1,
where
 (ℎ, 𝑘) is the center;
 𝑎 is the distance from
the center to a
vertex, and
 𝑏 is the distance from
the center to an
endpoint of the
conjugate axis.
Use this formula to find
the equation of a
hyperbola given its
center, 𝑎, and 𝑏 if the
transverse axis is
horizontal.

47. Key Formulas
47
Concept Formula Description
Equation of a
Hyperbola in Standard
Form
𝑦 − 𝑘 2
𝑎2 −
𝑥 − ℎ 2
𝑏2 = 1,
where
 (ℎ, 𝑘) is the center;
 𝑎 is the distance from
the center to a
vertex, and
 𝑏 is the distance from
the center to an
endpoint of the
conjugate axis.
Use this formula to find
the equation of a
hyperbola given its
center, 𝑎, and 𝑏 if the
transverse axis is
vertical.

48. Challenge Yourself
48
48
A nuclear power plant has a cooling tower
that is hyperboloid in shape. The width of
the slimmest part of the cooling tower is 60
meters and is 85 meters from the ground.
The diameter of the base of the tower is 120
meters, while the diameter of the top is 96
meters. What is the height of the tower?

49. Photo Credits
49
● Slides 2 and 3: Navigation boat Engineer Matusevich by Torin is licensed under CC BY-SA 3.0 via
Wikimedia Commons.
● Slide 13: Loran C Navigator by Morn the Gorn is licensed under CC BY-SA 3.0 via Wikimedia
Commons.
● Slides 17 to 20: Cooling towers of Dukovany Nuclear Power Plant in Dukovany, Třebíč District by
Jiří Sedláček is licensed under CC BY-SA 4.0 via Wikimedia Commons.

50. Bibliography
50
Barnett, Raymond, Michael Ziegler, Karl Byleen, and David Sobecki. College Algebra with Trigonometry. Boston:
McGraw Hill Higher Education, 2008.
Bittinger, Marvin L., Judith A. Beecher, David J. Ellenbogen, and Judith A. Penna. Algebra and Trigonometry:
Graphs and Models. 4th ed. Boston: Pearson/Addison Wesley, 2009.
Blitzer, Robert. Algebra and Trigonometry. 3rd ed. Upper Saddle River, New Jersey: Pearson/Prentice Hal, 2007.
Bourne, Murray. 6. The Hyperbola. Interactive Mathematics. Accessed from https://www.intmath.com/plane-
analytic-geometry/6-hyperbola.php. March 26, 2020.
Larson, Ron. College Algebra with Applications for Business and the Life Sciences. Boston: MA: Houghton Mifflin,
2009.
OpenStax College. Solving Applied Problems Involving Hyperbolas. Lumen Learning. Accessed from
https://courses.lumenlearning.com/ivytech-collegealgebra/chapter/solving-applied-problems-involving-
hyperbolas/. March 26, 2020.
Simmons, George F. Calculus with Analytic Geometry. 2nd ed. New York: McGraw-Hill, 1996.