11.8 Polar Equations of Conic Sections

(skip 11.7)

We know from Chapter 10, x2 + y2 = 64 is an equation of a circle and can be written as r = 8 in polar form.

We can write polar forms of equations for parabolas, ellipses, & hyperbolas (not circles) that have a focus at the pole and a directrix parallel or perpendicular to the polar axis.

* d is directrix and e is eccentricity

(we can derive equations using a general definition)

* if e = 1 parabola, if e > 1 hyperbola, if 0 < e < 1 ellipse

( , ) distance from point to focus

( , ) distance from point to directrix

d P Fe

d P l Often useful to use a

vertex for the point

1 cos

1 cos

3 and

1 sin 2 2

3 and

1 sin 2 2

edr

e

edr

e

edr

e

edr

e

PolarEquation

Directrix(d > 0 always) Axis To determine

vertices, let:

x = – d Horizontal θ = 0 and π

x = d Horizontal θ = 0 and π

y = – d Vertical

y = d Vertical

opp of directrixvalues that give trig function = 1

18

3 6cos18 6

3(1 2cos ) 1 2cos

6 6 6 66 2

1 2cos0 1 2 1 2cos 1 2

r

r

r r

Ex 1) Identify the conic with equation . Find vertices & graph.

let θ = 0:

e = 2 ed = 62d = 6 d = 3

directrix: x = –3

(–6, 0) (2, π)

let θ = π:

1 focus ALWAYS at pole!

Note: directrix not necessarily in middle

hyperbola

12

1 1 1 12 2 2 2

32 2

2

4 4sin2

4(1 sin ) 1 sin

1

1 sin 2 4 1 sin 0

1,

4 2

r

r

r r

let θ = :

e = 1 ed = ½ 1·d = ½ d = ½

directrix: y = ½

let θ = :

1 focus ALWAYS at pole!

parabola

1

–1

Try on your own:Ex 2) Identify the conic with equation . Find vertices & graph.

2 3

2

23

2 23 3

(3) 2 3 6

1 sin 1 sin 1 sin 3 3 2sin

edr

e

0

*We can also work backwards to find an equation.Ex 3) Find a polar equation for the conic with the given characteristic. a) Focus at the pole; directrix: y = –3; eccentricity:

form:

23

b) The vertices are (2, 0) and (8, π). Find eccentricity & identify the conic.

draw a sketch!

VV

C

F

1 focus ALWAYS at pole!center focus

center vertex3

5

ce

a

e

ellipse

to find d:

103

103

10 16 163 3 3

3 165 3

35

dist from V to F 2 3

dist from V to directrix dist 53(dist) 10 dist

(2,0) ,0

5 16

1 cos 5 5 3cos

e

V d

d

center halfway between vertices

Homework

#1109  Pg 586 #1, 5, 9, 13, 15, 17, 23, 25, 27, 29