Lesson 9.3Hyperbolas

Hyperbola

Set of all points where the difference between the distances to two fixed points (foci) is a positive constant.

20 cm12 cm

15 cm7cm

12 cm 4cm

10cm 2cm

FocusFocus

Other parts of a hyperbola:

Transversal axis can run vertically

Center

Axis (transverse)

Vertices

Foci

Equation of a Hyperbola

Similar to ellipse

Similarities/Differences:

•Subtraction between x2 and y2 terms

•Variable above a determines the direction of axis

•a is still the distance from center to a vertex

•c is still the distance from center to a focus

•c is now larger so…

12

2

2

2

b

y

a

x

Horizontal axis

12

2

2

2

b

x

a

y

Vertical axis

222 bac

Example

Find the standard form of the equation with foci (-1, 2) and (5, 2) and vertices (0, 2) and (4, 2).

Question: What is b in an hyperbola?

It is still a distance from the center in the opposite direction of the axis

How it applies to a parabola has to do with a new part unique to hyperbolas.

Asymptotes

Lines that bound the hyperbola

Pass through the diagonals of a rectangle with dimensions 2a and 2b

(h, k)(h - a, k) (h + a, k)

(h, k + b)

(h, k - b)

b is the distance from center to edge of rectangle along conjugate axis

Conjugate axis

Asymptotes

Equations for Asymptotes

hxa

bky hx

b

aky

Horizontal hyperbola (transverse axis)

b is up/down – a is left/right

slope is rise over run →a

b

Vertical hyperbola (transverse axis)

a is up/down – b is left/right

slope is rise over run →b

a

Example

Sketch the graph of 4x2 – y2 = 16, include the asymptotes.

Example

Find the standard form of the hyperbola with vertices (3, -5), (3, 1) and asymptotes

42

82

xy

xy

Eccentricity of an Hyperbola

where e > 1 (since c is larger than a)

large e → flatter curve

e close to 1 → more curved - pointed

a

ce

Problem Set 9.3