Post on 19-Jan-2016
Lesson8.5
Polar Equations of Conics
Precalculus
2
2 2
2
1. Solve for . (4, ) ( , )
2. Solve for . (3, 5 /3)=( 3, ), 2 2
3. Find the focus and the directrix of the parabola.
12
Find the focus and the vertices of the conic.
4. 1 16 9
5. 9
r r
x y
x y
x
2
1 16
y
Quick Review
4r ( , )r 53
4
3
2 4x py12 4p
3p
Focus: (0,3)
Directrix: y p 3y
16 4a 9 3b
16 9 5c Focus: ( ,0)c( 5,0)
Vertices: ( ,0)a( 4,0)
16 4a 9 3b 2 2 2a b c
Focus: (0, )c(0, 7)Vertices: (0, )a(0, 4)
16 9 7c
2 2 2c a b
What you’ll learn about
Eccentricity RevisitedWriting Polar Equations for ConicsAnalyzing Polar Equations of ConicsOrbits Revisited
… and whyYou will learn the approach to conics used by astronomers.
Focus-Directrix Definition Conic Section
The coordinates ( , ) and ( ', ') based on parallel sets of axes are
related by either of the following :
' and ' or ' and ' .
x y x y
x x h y y k x x h y y k translations formulas
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.)
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 where e is a
constant and the eccentricity of the conic.
Moreover, the conic is a hyperbola if e > 1,
a parabola if e = 1,
an ellipse if e < 1.
and ,PF
e PF e PDPD
The Geometric Structure of a Conic Section
A Conic Section in the Polar Plane
Three Types of Conics for r =
ke/(1+ecosθ)
x
y Directrix
DP
F(0,0)
x = k
1PF
ePD
Ellipse
x
y
Directrix
DP
F(0,0)
x = k
1PF
ePD
Parabola
x
y Directrix
DP
F(0,0)
x = k
1PF
ePD
Hyperbola
Polar Equations for Conics
Two standard orientations of a conic in the polar plane are as follows.
1 cos
ker
e
x
y
Directrix x = k
Focus at pole
1 cos
ker
e
x
y
Directrix x = k
Focus at pole
Polar Equations for Conics
The other two standard orientations of a conic in the polar plane are as follows.
1 sin
ker
e
x
y
Directrix y = k
Focus at pole
1 sin
ker
e
x
y
Directrix y = k
Focus at pole
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 3.x
4Setting and 3 in yields
5 1 cos
kee k r
e
3 4 / 5
1 4 / 5 cosr
12
5 4cosr
Example Identifying Conics from Their Polar Equations
Determine the eccentricity, the type of conic,
6 and the directrix.
3 2cosr
2The eccentricity is which means the conic eli ls ian
3pse.
Divide the numerator and the denominator by 3.
2
1 (2 / 3)cosr
2The numerator 2 , so 3 and the directrix is 3.
3ke k k x
Note, the sign in the denominator dictates the sign of the directrix.
Example Writing a Conic Section in Polar Form
Find a polar equation of the parabola with its focus at the pole
and directix 2.x
( )(2)
1
1
1( )cosr
Parabola eccentricity: 1e
Vertical Directrix: Use 21 cos
ekr k
e
Directrix is 2 units to the right of the vertex,
and is vertical:
Find a polar equation of the parabola with its focus at the pole
and vertex (2,0). Parabola eccentricity: 1e
of the ordered pair is 2, so 4 x k
( )( )
1
1
1( )c s
4
or
4
1 cosr
Example Writing a Conic Section in Polar Form
Find a polar equation of a conic section with the focus at the pole
6 and directix 2csc .e r
2y 2
sinr
sin 2r
( )( )
1
6
6( )s n
2
ir
12
1 6sinr
Recall
1 sin
ker
e
Example Writing a Conic Section in Polar Form
Find a polar equation for the ellipse with a focus at the pole
3and the given end-points of its major axis 1, and 3,
2 2
0
pi/2
1
2
cea
Recall
1 sin
ker
e
( )( )1
1 ( )
1/ 2
sin1 (/ 2 / 2)
k
1
1
0
(1
5
)
.
0.5
k
3k 3
2 sinr
/ 2
1/ 2 sin
3
1r
(0, 1)midpt
Homework:
Text pg683 Exercises
# 4-40 (intervals of 4)