pre-cal 40s may 25, 2009
DESCRIPTION
The hyperbola.TRANSCRIPT
12.3 Hyperbolas
The Anatomy of the Hyperbola
What does a "sonic boom" have to do with hyperbolas?
sonicboomplane_navy.jpg
Throw 2 stones in a pond. The resulting concentric ripples meet in a hyperbola shape.
Casting by flickr user harry harris
Hyperbolic Light
Sheffield Cooling Towers (HDR) by flickr user Paul Denton Cocker
Hyperbolic Cooling Towers
Generating Hyperbolas HOMEWORK
Constructing Hyperbolas
ASYMPTOTES
TRANSVERSEAXIS
CONJUGATEAXIS
VERTEX VERTEX
The Anatomy of The Hyperbola
O is the centre, it has coordinates (h, k).
P
a bc
cB1
B2
A1 A2
F1F2 O
The Anatomy of The Hyperbola
O is the centre, it has coordinates (h, k).A A is the length of the transverse axis. It's length is 2a.
21
P
a bc
cB1
B2
A1 A2
F1F2 O
The Anatomy of The Hyperbola
O is the centre, it has coordinates (h, k).A A is the length of the transverse axis. It's length is 2a.
21 B B is the length of the conjugate axis. It's length is 2b.
21
P
a bc
cB1
B2
A1 A2
F1F2 O
The Anatomy of The Hyperbola
O is the centre, it has coordinates (h, k).A A is the length of the transverse axis. It's length is 2a.
21 B B is the length of the conjugate axis. It's length is 2b.
21
F & F are called the foci. They are c units from the centre.
21
P
a bc
cB1
B2
A1 A2
F1F2 O
The Anatomy of The Hyperbola
O is the centre, it has coordinates (h, k).A A is the length of the transverse axis. It's length is 2a.
21 B B is the length of the conjugate axis. It's length is 2b.
21
PF & PF are the focal radii of the hyperbola.
21F & F are called the foci. They are c units from the centre.
21
P
a bc
cB1
B2
A1 A2
F1F2 O
The Anatomy of The Hyperbola
O is the centre, it has coordinates (h, k).A A is the length of the transverse axis. It's length is 2a.
21 B B is the length of the conjugate axis. It's length is 2b.
21
OA = OA is the length of the semitransverse axis with length a.
1 2
PF & PF are the focal radii of the hyperbola.
21F & F are called the foci. They are c units from the centre.
21
P
a bc
cB1
B2
A1 A2
F1F2 O
The Anatomy of The Hyperbola
O is the centre, it has coordinates (h, k).A A is the length of the transverse axis. It's length is 2a.
21 B B is the length of the conjugate axis. It's length is 2b.
21
OA = OA is the length of the semitransverse axis with length a.
1 2 A & A are called the vertices of the hyperbola.
21
PF & PF are the focal radii of the hyperbola.
21F & F are called the foci. They are c units from the centre.
21
P
a bc
cB1
B2
A1 A2
F1F2 O
The Anatomy of The Hyperbola
O is the centre, it has coordinates (h, k).A A is the length of the transverse axis. It's length is 2a.
21 B B is the length of the conjugate axis. It's length is 2b.
21
OA = OA is the length of the semitransverse axis with length a.
1 2
OB = OB is the length of the semiconjugate axis with length b.
1 2
A & A are called the vertices of the hyperbola.
21
PF & PF are the focal radii of the hyperbola.
21F & F are called the foci. They are c units from the centre.
21
P
a bc
cB1
B2
A1 A2
F1F2 O
The Anatomy of The Hyperbola
O is the centre, it has coordinates (h, k).A A is the length of the transverse axis. It's length is 2a.
21 B B is the length of the conjugate axis. It's length is 2b.
21
OA = OA is the length of the semitransverse axis with length a.
1 2
OB = OB is the length of the semiconjugate axis with length b.
1 2
A & A are called the vertices of the hyperbola.
21
B & B are called the endpoints of the conjugate axis.
21
PF & PF are the focal radii of the hyperbola.
21F & F are called the foci. They are c units from the centre.
21
P
a bc
cB1
B2
A1 A2
F1F2 O
The Standard Form for the Equation of a Hyperbola
Horizontal Orientation Vertical Orientation
Similarities Differences
• Both equal one• have (x-h)^2 and (y-k)^2• have a^2 and b^2 in denominator.• b^2 denominator in negative term.• a^2 denominator in positive term.
• a^2 with x^2 on left, a^2 with y^2 on right.• b^2 with y^2 on left, a^2 with x^2 on right• when y (+) is vertical, when y(-) is horizontal. • when x (-) is vertical, x(+) is horizontal.
a bO
a
bO
Horizontal Hyperbola Vertical Hyperbola
The Pythagorean Property
b2c2 a2+=
O A1
B1
A2
ca F1F2
b
B2
c
Horizontal Hyperbola Vertical Hyperbola
Conics Animations Sourcehttp://tinyurl.com/conicAni
Conics Animations Source
http://tinyurl.com/conicAni
Conics Animations Source
http://tinyurl.com/conicAni
Conics Animations Source
http://tinyurl.com/conicAni
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.
(iii) Sketch a graph of the hyperbola.HOMEWORK