Pre-Cal 40S May 25, 2009

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


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



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




21

F & F are called the foci. They are c units from the centre.

21

P

a bc

cB1

B2

A1 A2

F1F2 O




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




21

OA = OA is the length of the semitransverse axis with length a.

1 2



21

P

a bc

cB1

B2

A1 A2

F1F2 O




21


1 2 A & A are called the vertices of the hyperbola.

21



21

P

a bc

cB1

B2

A1 A2

F1F2 O




21


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



21

P

a bc

cB1

B2

A1 A2

F1F2 O




21


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



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

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

Pre-Cal 40S May 25, 2009

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Transcript of Pre-Cal 40S May 25, 2009