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Polar Equations of Conics

Concept:

The equations of conic sections in polar form take an especially simple and unified form when a focus is at the pole and an axis is parallel or perpendicular to the ray θ = 0°.

r = pe
(1 - e cos θ)
r = pe
(1 + e cos θ)
r = pe
(1 - e sin θ)
r = pe
(1 + e sin θ)

Changing the sign in the denominator or changing the trigonometric function used in the denominator to the sine function changes the orientation of the graph, but not its shape or size.

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Opera
ellipse parabola Hyperbola
0<e<1 e=1 e>1
Click on the picture of the conic section you want to investigate.
latus rectum:

eccentricity:
1
+ - equation: r =
trig function: sine cosine
1
1θ
vertex: (, ) focal length: axis: = 0 directrix: =
latus rectum:

eccentricity:

+ - equation: r =
trig function: sine cosine
1
1 θ
vertices: (, ) and (,)
foci: (0,0) and (,)
major axis:
covertices: (,) and (,)
center: (,)
minor axis:
directrices: = and =
latus rectum:

eccentricity:

+ - equation: r =
trig function: sine cosine
1
1 θ
vertices: (, ) and (,) foci: (0,0) and (,) transverse axis:
covertices: (,) and (,) center: (,) conjuate axis:
directrices: = and = asymptotes: = x and = x

INSTRUCTIONS

EXPLORATIONS

PRACTICE QUESTIONS

1 2 3 4 5 6
1. Which parabola has the directrix
y = -8?
2. The ellipse with polar form
r =
3.6
1+0.64 cosθ
has what point
as a vertex?



3. What is the non-polar focus of the
hyperbola r =
6.5
1-2.5 cosθ
?

4. What is the vertex of the parabola
r =
5
1+ cosθ
?

5. Find the length of the minor axis of
the ellipse r =
1.8
1-0.8 sinθ
 .

6. The hyperbola r =
9.8
1+4.1 sinθ
has
which of these oblique lines as an asymptote?



x can be any number from to

t can be any number from to