The document defines and explains hyperbolas through the following key points: 1. A hyperbola is the set of points where the absolute difference between the distance to two fixed points (foci) is a constant. 2. Key parts of a hyperbola include vertices, foci, transverse axis, and conjugate axis. 3. The standard equation of a hyperbola is (x2/a2) - (y2/b2) = 1 4. Examples are worked through to graph specific hyperbolas using their equations.

1. 10.3 Hyperbolas
John 17:3 "And this is eternal life, that they know you the
only true God, and Jesus Christ whom you have sent."

2. Another way to define
the hyperbola is this:
A Hyperbola is the set
of points in the plane,
the difference of whose
distances from two
fixed points (the Foci),
is a constant.
use K on keyboard to Pause/Play

3. Another way to define
the hyperbola is this:
A Hyperbola is the set
of points in the plane,
the difference of whose
distances from two
fixed points (the Foci),
is a constant.
use K on keyboard to Pause/Play

5. F2 F1

6. Vertex Vertex
F2 F1

7. Central Box
Vertex Vertex
F2 F1

8. Central Box
Vertex Vertex
F2 F1
Asymptotes

9. Central Box
Vertex Vertex
F2 F1
Asymptotes
The segment joining V1 and V2 is the Transverse Axis.

11. To derive the equation for a hyperbola ... start with
PF2 − PF1 = 2a

12. To derive the equation for a hyperbola ... start with
PF2 − PF1 = 2a
It is very similar to what we did with the ellipse, so we
won’t go through all of that again. The equation is:

13. To derive the equation for a hyperbola ... start with
PF2 − PF1 = 2a
It is very similar to what we did with the ellipse, so we
won’t go through all of that again. The equation is:
2 2
x y
2
− 2 =1
a b

14. A bit more detail ...

15. A bit more detail ...
V1V2 :Transverse Axis
B1 B2 :Conjugate Axis

16. A bit more detail ...
V1V2 :Transverse Axis
B1 B2 :Conjugate Axis
2 2
x y
2
− 2 = 1 opens left − right
a b

17. A bit more detail ...
V1V2 :Transverse Axis
B1 B2 :Conjugate Axis
2 2 2 2
x y y x
2
− 2 = 1 opens left − right 2
− 2 = 1 opens up − down
a b a b

18. To sketch a hyperbola:

19. To sketch a hyperbola:
1. Draw the central box

20. To sketch a hyperbola:
1. Draw the central box
2. Draw in the asymptotes

21. To sketch a hyperbola:
1. Draw the central box
2. Draw in the asymptotes
3. Plot the vertices and foci

22. To sketch a hyperbola:
1. Draw the central box
2. Draw in the asymptotes
3. Plot the vertices and foci
4. Draw the hyperbola

23. 2 2
1. Discuss and graph: x y
− =1
16 25

24. 2 2
1. Discuss and graph: x y
− =1
16 25
2 2 2
a=4 b=5 c = a +b
c = 41 ≈ 6.4

25. 2 2
1. Discuss and graph: x y
− =1
16 25
2 2 2
a=4 b=5 c = a +b
c = 41 ≈ 6.4
Vert. : ( ± 4,0 )
Foci : ( ± 6.4,0 )
TA : 8
CA : 10
5
Asy. : y = ± x
4

26. 2. Discuss and graph: 2 2
4x − y + 4 = 0

27. 2. Discuss and graph: 2 2
4x − y + 4 = 0
2 2
4x − y = −4

28. 2. Discuss and graph: 2 2
4x − y + 4 = 0
2 2
4x − y = −4
2 2
y x
− =1
4 1

29. 2. Discuss and graph: 2 2
4x − y + 4 = 0
2 2
4x − y = −4
2 2
y x
− =1
4 1
2 2 2
a=2 b =1 c = a +b
c = 5 ≈ 2.2

30. 2. Discuss and graph: 2 2
4x − y + 4 = 0
2 2
4x − y = −4
2 2
y x
− =1
4 1
2 2 2
a=2 b =1 c = a +b
c = 5 ≈ 2.2
Vert. : ( 0, ± 2 )
Foci : ( 0, ± 2.2 )
TA : 4
CA : 2
x
Asy. : y = ±
2

31. HW #6
“The difference between the possible and the impossible
lies in a man’s determination.”
Tommy Lasorda