polar equations of conics
DESCRIPTION
Precalculus. Lesson 8.5. Polar Equations of Conics. Quick Review. 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. - PowerPoint PPT PresentationTRANSCRIPT
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)