Copyright © 2011 Pearson, Inc.
8.5
Polar Equations
of Conics
Copyright © 2011 Pearson, Inc. Slide 8.5 - 2
What you’ll learn about
Eccentricity Revisited
Writing Polar Equations for Conics
Analyzing Polar Equations of Conics
Orbits Revisited
… and why
You will learn the approach to conics used by
astronomers.
Copyright © 2011 Pearson, Inc. Slide 8.5 - 3
Focus-Directrix Definition Conic
Section
A conic section is the set of all points in a
plane whose distances from a particular point
(the focus) and a particular line (the directrix)
in the plane have a constant ratio. (We assume
that the focus does not lie on the directrix.)
Copyright © 2011 Pearson, Inc. Slide 8.5 - 4
Focus-Directrix Eccentricity
Relationship
If P is a point of a conic section, F is the conic's focus,
and D is the point of the directrix closest to P, then
e PF
PD and PF e PD, where e is a constant and
the eccentricity of the conic. Moreover, the conic is
g a hyperbola if e 1,
g a parabola if e 1,
g an ellipse if e 1.
Copyright © 2011 Pearson, Inc. Slide 8.5 - 5
The Geometric Structure of a Conic
Section
Copyright © 2011 Pearson, Inc. Slide 8.5 - 6
A Conic Section in the Polar Plane
Copyright © 2011 Pearson, Inc. Slide 8.5 - 7
Three Types of Conics for r =
ke/(1+ecosθ)
Copyright © 2011 Pearson, Inc. Slide 8.5 - 8
Polar Equations for Conics
Copyright © 2011 Pearson, Inc. Slide 8.5 - 9
Example Writing Polar Equations of
Conics
Given that the focus is at the pole, write a polar equation
for the conic with eccentricity 4/5 and directrix x 3.
Copyright © 2011 Pearson, Inc. Slide 8.5 - 10
Example Writing Polar Equations of
Conics
Setting e 4 / 5 and k 3 in r ke
1 ecos yields
r 3 4 / 5
1 4 / 5 cos
12
5 cos
Given that the focus is at the pole, write a polar equation
for the conic with eccentricity 4/5 and directrix x 3.
Copyright © 2011 Pearson, Inc. Slide 8.5 - 11
Example Identifying Conics from
Their Polar Equations
Determine the eccentricity, the type of conic,
and the directrix. r 6
3 2cos
Copyright © 2011 Pearson, Inc. Slide 8.5 - 12
Example Identifying Conics from
Their Polar Equations
Divide the numerator and the denominator by 3.
r 2
1 (2 / 3)cos
The eccentricity is 2/3 which means the conic is an ellipse.
The numerator ke 2 (2 / 3)k, so k 3
and the directrix is y 3.
Determine the eccentricity, the type of conic,
and the directrix. r 6
3 2cos
Copyright © 2011 Pearson, Inc. Slide 8.5 - 13
Example Analyzing a Conic
Analyze the conic section given by the equation
r 6
3 2cos
Include in the analysis the values of e, a, b, and c.
Copyright © 2011 Pearson, Inc. Slide 8.5 - 14
Example Analyzing a Conic
Divide the numerator and the
denominator by 3 yields
r 6
1 2cos
The eccentricity is e 2, and
thus the conic is a
hyperbola, shown here.
r
6
3 2cos
Copyright © 2011 Pearson, Inc. Slide 8.5 - 15
Example Analyzing a Conic
The vertices have polar coordinates (2, 0) and (Ğ6, ).
So 2a 6 2 4, and thus a 2
The vertex (2, 0) is 2 units to the
right of the pole, the pole is the
focus of the hyperbola,
so c a 2, and thus c 4.
r
6
3 2cos
Copyright © 2011 Pearson, Inc. Slide 8.5 - 16
Example Analyzing a Conic
To find b, use Pythagorean relation:
b c2 a2 16 4 12 2 3
With all the information, we can
write the Cartesian equation of
the hyperbola:
x 4 2
4
y2
12 1
Copyright © 2011 Pearson, Inc. Slide 8.5 - 17
Semimajor Axes and Eccentricities of
the Planets
Copyright © 2011 Pearson, Inc. Slide 8.5 - 18
Ellipse with Eccentricity e and
Semimajor Axis a
21
1 cos
a er
e
Copyright © 2011 Pearson, Inc. Slide 8.5 - 19
Quick Review
1. Solve for r. (4,) (r, )
2. Solve for . (3, 5 /3)=( 3,), 2 2
3. Find the focus and the directrix of the parabola.
x2 12y
Find the focus and the vertices of the conic.
4. x2
16
y2
9 1
5. x2
9
y2
16 1
Copyright © 2011 Pearson, Inc. Slide 8.5 - 20
Quick Review Solutions
1. Solve for r. (4,) (r, ) 4
2. Solve for . (3, 5 /3)=( 3,), 2 2 4 / 3
3. Find the focus and the directrix of the parabola.
x2 12y (0,3); y 3
Find the focus and the vertices of the conic.
4. x2
16
y2
9 1 (5,0); ( 4,0)
5. x2
9
y2
16 1 (0, 7); (0, 4)