polar equations of conics it’s a whole new ball game in section 8.5a…

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Polar Polar Equations of Equations of

ConicsConicsIt’s a whole new ball It’s a whole new ball game in Section 8.5a…game in Section 8.5a…

Focus-Directrix Definition:Conic Section

A conic section is the set of all points in a plane whosedistances from a particular point (the focus) and a particularline (the directrix) in the plane have a constant ratio. (Weassume that the focus does not lie on the directrix.)

Here, we are generalizing the focus-directrixHere, we are generalizing the focus-directrixdefinition given for parabolas in section 8.1 todefinition given for parabolas in section 8.1 to

apply to all three of our conic sections!!!apply to all three of our conic sections!!!

Focus-Directrix Definition:Conic Section

Conicsection P

F

Focus

Vertex

Focalaxis

Directrix

D

Focal Axis – line passingthrough the focus and perp.to the directrix

Vertex – point where theconic intersects its axis

Eccentricity (e) – theconstant ratio PF

PDA parabola has one focus and one directrix…

Ellipses and hyperbolas have two focus-directrix pairs…

Focus-Directrix-Eccentricity Relationship

If P is a point of a conic section, F is the conic’s focus, and Dis the point of the directrix closest to P, then

PFe

PD and PF e PD

where the constant e is the eccentricity of the conic.Moreover, the conic is

• a hyperbola if e > 1,

• a parabola if e = 1,

• an ellipse if e < 1.

Writing Polar Equations for Conics

Our previous definition for conics works best in combinationwith polar coordinates……………..so remind me:

Pole: the origin

Polar Axis: the x-axis

Pole Polar Axis

,P r

r

To obtain a polar equation for a conic section, we position thepole at the conic’s focus and the polar axis along the focal axiswith the directrix to the right of the pole…

Writing Polar Equations for Conics

Conicsection

Directrix

Focus atthe pole

,P r r

D

F cosr

cosk r

x k

We call the distance fromthe focus to the directrix k

PF rcosPD k r

our previous equation

PF e PD becomes

cosr e k r

Writing Polar Equations for Conics

Conicsection

Directrix

Focus atthe pole

,P r r

D

F cosr

cosk r

x k

Solve this for r :

cosr e k r cosr ke re

cosr re ke 1 cosr e ke

1 cos

ker

e

Writing Polar Equations for Conics

This one equation can produce all types of conic sections.

1 cos

ker

e

If 1PF

ePD

Ellipse!Ellipse!F(0, 0)

P D

x = k

Directrix

Writing Polar Equations for Conics

This one equation can produce all types of conic sections.

1 cos

ker

e

If 1PF

ePD

Parabola!Parabola!F(0, 0)

P D

x = k

Directrix

Writing Polar Equations for Conics

This one equation can produce all types of conic sections.

1 cos

ker

e

If 1PF

ePD

Hyperbola!Hyperbola!F(0, 0)

P D

x = k

Directrix

A Fun Calculator “Exploration”

Set your grapher to Polar and Dot graphing modes, and toRadian mode. Using k = 3, an xy window of [–12, 24] by[–12, 12], 0min = 0, 0max = 2 , and 0step = /48, graph

1 cos

ker

e

for e = 0.7, 0.8, 1, 1.5, 3. Identify the type of conic sectionobtained for each e value.

Overlay the five graphs, and explain how changing the valueof e affects the graph.

Explain how the five graphs are similar and how they aredifferent.

Polar Equations for Conics

The four standard orientations of a conic in the polar plane areas follows.

1 cos

ker

e

Focus at pole

Directrixx = k

1 cos

ker

e

Focus at pole

Directrixx = –k

Polar Equations for Conics

The four standard orientations of a conic in the polar plane areas follows.

1 sin

ker

e

Focusat pole

Directrix y = k

1 sin

ker

e

Focusat pole

Directrix y = –k

Practice ProblemsGiven that the focus is at the pole, write a polar equation for thespecified conic, and graph it.

1 cos

ker

e

Eccentricity e = 3/5, Directrix x = 2

GeneralEquation:

Substitute in thegiven info:

2 3 5

1 3 5 cosr

Multiply numeratorand denominator by 5:

6

5 3cosr

Now, how do we graph this conic???

(by hand and by calculator)

Practice ProblemsGiven that the focus is at the pole, write a polar equation for thespecified conic, and graph it.

Eccentricity e = 1, Directrix x = –2

2

1 cosr

The graph???

Practice ProblemsGiven that the focus is at the pole, write a polar equation for thespecified conic, and graph it.

Eccentricity e = 3/2, Directrix y = 4

12

2 3sinr

The graph???

Practice ProblemsDetermine the eccentricity, the type of conic, and the directrix.

6

2 3cosr

3

1 1.5cosr

Divide numeratorand denominator

by 2:

e = 1.5 Hyperbola!!!

ke = 3 k = 2

Directrix: x = 2

Verify all of this graphically???

Practice ProblemsDetermine the eccentricity, the type of conic, and the directrix.

6

4 3sinr

1.5

1 0.75sin

e = 0.75 Ellipse!!!

k = 2 Directrix: y = –2

Verify all of this graphically???

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