HyperbolasHyperbolasSec. 8.3aSec. 8.3a

Definition: HyperbolaDefinition: Hyperbola

A hyperbola is the set of all points in a plane whosedistances from two fixed points in the plane have aconstant difference. The fixed points are the foci of thehyperbola. The line through the foci is the focal axis.The point on the focal axis midway between the foci isthe center. The points where the hyperbola intersectsits focal axis are the vertices of the hyperbola.

How is this different from an ellipse???How is this different from an ellipse???

Definition: HyperbolaDefinition: Hyperbola

Center

Vertex Vertex FocusFocus

Focal Axis

Deriving the Equation of a HyperbolaDeriving the Equation of a Hyperbola

0

,P x y

x a

2 ,0F c

x a

1 ,0F c

Notice:

c a1 2 2PF PF a

2 1 2PF PF a Combining:

1 2 2PF PF a Distance Formula:

2 2 2 20 0 2x c y x c y a

Deriving the Equation of a HyperbolaDeriving the Equation of a Hyperbola

2 2 2 20 0 2x c y x c y a

2 22 22x c y a x c y

22 2 2 2 2 2 2 22 4 4 2x cx c y a a x c y x cx c y

2 2 2a x c y a cx

2 2 2 2 4 2 2 22 2a x cx c y a a cx c x 2 2 2 2 2 2 2 4 2 2 22 2a x a cx a c a y a a cx c x

Deriving the Equation of a HyperbolaDeriving the Equation of a Hyperbola

2 2 2 2 2 2 2 2c a x a y a c a

2 2 2 2 2 2 2 4 2 2 22 2a x a cx a c a y a a cx c x

2 2 2b c a Let

2 2 2 2 2 2b x a y a b 2 2a bDivide both sides by

2 2

2 21

x y

a b

Deriving the Equation of a HyperbolaDeriving the Equation of a Hyperbola

This equation is the standard form of the equation of a hyperbolacentered at the origin with the x-axis as its focal axis.

When the y-axis is the focal axis?2 2

2 21

y x

a b

Chord – segment with endpoints on the hyperbola

Transverse Axis – chord lying on the focal axis, connectingthe vertices (length = 2a)

2 2

2 21

x y

a b

Conjugate Axis – segment (length = 2b) that is perp. to thefocal axis and has the center of the hyperbola as its midpoint

Deriving the Equation of a HyperbolaDeriving the Equation of a Hyperbola

This equation is the standard form of the equation of a hyperbolacentered at the origin with the x-axis as its focal axis.

When the y-axis is the focal axis?2 2

2 21

y x

a b

Semitransverse Axis – the number “a”

2 2

2 21

x y

a b

Semiconjugate Axis – the number “b”

Deriving the Equation of a HyperbolaDeriving the Equation of a Hyperbola

The hyperbola

has two asymptotes, which can be found by replacing the “1”in the equation with a “0”:

2 2

2 20

x y

a b Solve for y

2 2

2 21

x y

a b

by x

a

Drawing Practice:Drawing Practice:Steps to sketching the hyperbolaSteps to sketching the hyperbola

2 2

2 21

x y

a b

Hyperbolas with Center (0, 0)Hyperbolas with Center (0, 0)

• Standard Equation

2 2

2 21

x y

a b

• Focal Axis x-axis

• Foci ,0c• Vertices ,0a• Semitrans. Axis a• Semiconj. Axis b• Pythagorean Relation

2 2 2c a b

• Asymptotesb

y xa

2 2

2 21

y x

a b

y-axis

0, c

0, aab2 2 2c a b

ay x

b

Hyperbolas with Center (0, 0)Hyperbolas with Center (0, 0)

,0c ,0c ,0a ,0a

by x

a

by x

a

2 2

2 21

x y

a b

Hyperbolas with Center (0, 0)Hyperbolas with Center (0, 0)

0, c

0,c

0, a

0,a

ay x

b

ay x

b

2 2

2 21

y x

a b

Guided PracticeGuided PracticeFind the vertices and the foci of the hyperbola

2 24 9 36x y

Standard Equation:2 2

19 4

x y Sketch the

hyperbola?

2 4b 2 9a

3,0

2 2 2 13c a b Vertices: 13,0Foci:

Guided PracticeGuided PracticeFind an equation of the hyperbola with foci (0, –3) and (0, 3)whose conjugate axis has length 4. Sketch the hyperbola andits asymptotes, and support your sketch with a grapher.

2 2

2 21

y x

a b General Equation:

c = 3 b = 2

a = 5

2 2

15 4

y x Standard Equation:

2

5 14

xy

The Sketch???

Let’s see some hyperbolas whose centers are not on the origin…

,h k

y k ,h c k

,h a k

,h c k

,h a k

by x h k

a

by x h k

a

Let’s see some hyperbolas whose centers are not on the origin…

,h k

x h

,h k c ,h k a

,h k a ,h k c

ay x h k

b

ay x h kb

Hyperbolas with Center (Hyperbolas with Center (hh, , kk))

• Standard Equation 2 2

2 21

x h y k

a b

• Focal Axis y k• Foci ,h c k• Vertices ,h a k• Semitransverse Axis a• Semiconjugate Axis b

• Pythagorean Relation2 2 2c a b

• Asymptotes by x h k

a

Hyperbolas with Center (Hyperbolas with Center (hh, , kk))

• Standard Equation 2 2

2 21

y k x h

a b

• Focal Axis x h• Foci ,h k c• Vertices ,h k a• Semitransverse Axis a• Semiconjugate Axis b

• Pythagorean Relation2 2 2c a b

• Asymptotes ay x h k

b

Guided PracticeGuided Practice

Find the standard form of the equation for the hyperbolawhose transverse axis has endpoints (–2, –1) and (8, –1),and whose conjugate axis has length 8.

Start with a diagram?

General Equation: 2 2

2 21

x h y k

a b

The center is the midpointof the transverse axis: , 3, 1h k

Guided PracticeGuided PracticeFind the standard form of the equation for the hyperbolawhose transverse axis has endpoints (–2, –1) and (8, –1),and whose conjugate axis has length 8.

Semitransverse Axis:

8 25

2a

Semiconjugate Axis:8

42

b

Specific Equation: 2 2

3 11

25 16

x y

Guided PracticeGuided PracticeFind the center, vertices, and foci of the given hyperbola.

2 22 5

19 49

x y

Center: 2,5 3a 7b 58c Vertices: 2 3,5 1,5 , 5,5

The graph?

Foci: 2 58,5 5.616,5 , 9.616,5

Guided PracticeGuided PracticeFind an equation in standard form for the hyperbola withtransverse axis endpoints (–2, –2) and (–2, 7), slope ofone asymptote 4/3. Start with a graph?

Center is the midpointof the transverse axis: 5

, 2,2

h k

Find a, the semi-transverse axis:9

2a

Asymptote slope is a/b:9 2 4

3b

27

8b

Guided PracticeGuided PracticeFind an equation in standard form for the hyperbola withtransverse axis endpoints (–2, –2) and (–2, 7), slope ofone asymptote 4/3.

General equation: 2 2

2 21

y k x h

a b

Plug in data: 2 2

5 2 21

81 4 729 64

y x