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)