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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???