Polar Equations of Conics It’s a whole new ball game in Section 8.5a…
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Transcript of Polar Equations of Conics It’s a whole new ball game in Section 8.5a…
![Page 1: Polar Equations of Conics It’s a whole new ball game in Section 8.5a…](https://reader035.fdocuments.in/reader035/viewer/2022062315/5697c01a1a28abf838ccf0b8/html5/thumbnails/1.jpg)
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…
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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!!!
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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…
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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.
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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…
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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
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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
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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
![Page 9: Polar Equations of Conics It’s a whole new ball game in Section 8.5a…](https://reader035.fdocuments.in/reader035/viewer/2022062315/5697c01a1a28abf838ccf0b8/html5/thumbnails/9.jpg)
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
![Page 10: Polar Equations of Conics It’s a whole new ball game in Section 8.5a…](https://reader035.fdocuments.in/reader035/viewer/2022062315/5697c01a1a28abf838ccf0b8/html5/thumbnails/10.jpg)
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
![Page 11: Polar Equations of Conics It’s a whole new ball game in Section 8.5a…](https://reader035.fdocuments.in/reader035/viewer/2022062315/5697c01a1a28abf838ccf0b8/html5/thumbnails/11.jpg)
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.
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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
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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
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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)
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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???
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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???
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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???
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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???