The hyperbola.

1. The Anatomy of
the Hyperbola
12.3 Hyperbolas

2. What does a quot;sonic boomquot;
have to do with hyperbolas?
sonicboomplane_navy.jpg

4. Throw 2 stones in a pond. The resulting concentric ripples
meet in a hyperbola shape.

5. Hyperbolic
Light
Casting by flickr
user harry harris

6. Hyperbolic
Cooling Towers
Sheffield Cooling Towers (HDR)
by flickr user Paul Denton Cocker

7. Generating Hyperbolas HOMEWORK

10. Constructing Hyperbolas
CONJUGATE
AXIS
VERTEX
VERTEX
TRANSVERSE
AXIS
ASYMPTOTES

11. The Anatomy of The Hyperbola P
B1
c F
F2 a b 1
A 1 O c A2
B2
O is the centre, it has coordinates (h, k).

12. The Anatomy of The Hyperbola P
B1
c F
F2 a b 1
A 1 O c A2
B2
O is the centre, it has coordinates (h, k).
A1A2 is the length of the
transverse axis. It's length is 2a.

13. The Anatomy of The Hyperbola P
B1
c F
F2 a b 1
A 1 O c A2
B2
O is the centre, it has coordinates (h, k).
A1A2 is the length of the B1 B2 is the length of the
transverse axis. It's length is 2a. conjugate axis. It's length is 2b.

14. The Anatomy of The Hyperbola P
B1
c F
F2 a b 1
A 1 O c A2
B2
O is the centre, it has coordinates (h, k).
A1A2 is the length of the B1 B2 is the length of the
transverse axis. It's length is 2a. conjugate axis. It's length is 2b.
F 1 & F 2 are called the foci. They
are c units from the centre.

15. The Anatomy of The Hyperbola P
B1
c F
F2 a b 1
A 1 O c A2
B2
O is the centre, it has coordinates (h, k).
A1A2 is the length of the B1 B2 is the length of the
transverse axis. It's length is 2a. conjugate axis. It's length is 2b.
F 1 & F 2 are called the foci. They PF1 & PF 2 are the focal
are c units from the centre. radii of the hyperbola.

16. The Anatomy of The Hyperbola P
B1
c F
F2 a b 1
A 1 O c A2
B2
O is the centre, it has coordinates (h, k).
A1A2 is the length of the B1 B2 is the length of the
transverse axis. It's length is 2a. conjugate axis. It's length is 2b.
F 1 & F 2 are called the foci. They PF1 & PF 2 are the focal
are c units from the centre. radii of the hyperbola.
OA1 = OA2 is the length of the
semitransverse axis with length a.

17. The Anatomy of The Hyperbola P
B1
c F
F2 a b 1
A 1 O c A2
B2
O is the centre, it has coordinates (h, k).
A1A2 is the length of the B1 B2 is the length of the
transverse axis. It's length is 2a. conjugate axis. It's length is 2b.
F 1 & F 2 are called the foci. They PF1 & PF 2 are the focal
are c units from the centre. radii of the hyperbola.
OA1 = OA2 is the length of the A1 & A 2 are called the
semitransverse axis with length a. vertices of the hyperbola.

18. The Anatomy of The Hyperbola P
B1
c F
F2 a b 1
A 1 O c A2
B2
O is the centre, it has coordinates (h, k).
A1A2 is the length of the B1 B2 is the length of the
transverse axis. It's length is 2a. conjugate axis. It's length is 2b.
F 1 & F 2 are called the foci. They PF1 & PF 2 are the focal
are c units from the centre. radii of the hyperbola.
OA1 = OA2 is the length of the A1 & A 2 are called the
semitransverse axis with length a. vertices of the hyperbola.
OB1 = OB2 is the length of the
semiconjugate axis with length b.

19. The Anatomy of The Hyperbola P
B1
c F
F2 a b 1
A 1 O c A2
B2
O is the centre, it has coordinates (h, k).
A1A2 is the length of the B1 B2 is the length of the
transverse axis. It's length is 2a. conjugate axis. It's length is 2b.
F 1 & F 2 are called the foci. They PF1 & PF 2 are the focal
are c units from the centre. radii of the hyperbola.
OA1 = OA2 is the length of the A1 & A 2 are called the
semitransverse axis with length a. vertices of the hyperbola.
OB1 = OB2 is the length of the B1 & B2 are called the endpoints
semiconjugate axis with length b. of the conjugate axis.

20. The Standard Form for the Equation of a Hyperbola
Horizontal Orientation Vertical Orientation
Similarities Differences
• Both equal one • a^2 with x^2 on left,
• have (x-h)^2 and (y-k)^2 a^2 with y^2 on right.
• have a^2 and b^2 in • b^2 with y^2 on left,
denominator. a^2 with x^2 on right
• b^2 denominator in negative • when y (+) is vertical,
term. when y(-) is horizontal.
• a^2 denominator in positive • when x (-) is vertical,
term. x(+) is horizontal.

21. Horizontal Hyperbola Vertical Hyperbola
a b a
O
O b

22. The Pythagorean Property
B1
b a c
F1
F2 A2 O c A 1
B2
2 2 2
c = a +b
Horizontal Hyperbola Vertical Hyperbola

23. Conics Animations Source
http://tinyurl.com/conicAni

24. Conics Animations Source
http://tinyurl.com/conicAni

25. Conics Animations Source
http://tinyurl.com/conicAni

26. Conics Animations Source
http://tinyurl.com/conicAni

27. For the hyperbola whose equation is given below.
(i) Write the equation in standard form
(ii) Determine the lengths of the transverse and conjugate axes,
the coordinates of the verticies and foci, and the equations of the
asymptotes. HOMEWORK
(iii) Sketch a graph of the hyperbola.