May 11, 2015
Name: ___________________
Prob. 8.4
Use the discriminant to identify the type of conic. Then rotate
the coordinate axes to eliminate the xy term. Write and graph
the transformed equation.
5x2 - 2xy + 5y2 - 12 = 0
Discriminant = ______
Type of Conic = _________
Angle of Rotation = _______
Equ. in x'y' System: _________________
Vertices in x'y' System: _____________
Vertices in xy System: _____________
May 11, 2015
Name: ___________________
Prob. 8.4
Use the discriminant to identify the type of conic. Then rotate
the coordinate axes to eliminate the xy term. Write and graph
the transformed equation.
5x2 - 2xy + 5y2 - 12 = 0
Discriminant = ______
Type of Conic = _________
Angle of Rotation = _______
Equ. in x'y' System: _________________
Vertices in x'y' System: _____________
Vertices in xy System: _____________
May 11, 2015
(8.5) Polar Equations of Conics
Objective: To write and analyze polar equations of conics and
write their equivalent rectangular equation.
Why: Polar equations of conics are used by astronomers.
May 11, 2015
Obj: To write and analyze polar equations of conics and
write their equivalent rectangular equation.
Polar Equations of Conics (with one focus at pole)
Defn: all pts in which the distance from a fixed
point (focus) is in constant ratio (eccentricity)
to the distance from a fixed line (directrix)
Parabola: e=1
P
Ellipse: e < 1
P
D
F
PF = 1
PD
Hyperbola: e > 1
P
D
F
PF < 1
PD
b2 = a2(1-e2)
F
PF > 1
PD
b2 = a2(e2-1)
D
e= PF
PD
May 11, 2015
Obj: To write and analyze polar equations of conics and
write their equivalent rectangular equation.
x=k
P(r,θ)
F (0,0)
k = distance from F to directrix
e= PF
PD
D
May 11, 2015
Obj: To write and analyze polar equations of conics and
write their equivalent rectangular equation.
Two Forms of Polar Equations of a Conic
1. r =
ke
1+ ecosO
Vertical Directrix
e > 0,
+ (left or down)
- (right or up)
2. r = ke
1+ esinO
Horizontal Directrix
k = distance between focus(pole) and directrix
May 11, 2015
Obj: To write and analyze polar equations of conics and
write their equivalent rectangular equation.
Find a polar equation for the conic with a focus at the pole
and the given eccentricity and directrix. Identify the
conic.
1. e = 3/2, and directrix y = 2
2. e = 1, and directrix x = -2
May 11, 2015
Obj: To write and analyze polar equations of conics and
write their equivalent rectangular equation.
1. Determine the eccentricity, type of conic, and the
directrix:
r=
15
3 - 2cosO
May 11, 2015
2. r =
32
3 +5 sinO
Obj: To write and analyze polar equations of conics and
write their equivalent rectangular equation.
May 11, 2015
Obj: To write and analyze polar equations of conics and
write their equivalent rectangular equation.
Analyze the conic section given by the equation
below. Include values of e, a, b, and c. Determine the
equivalent Cartesian equation.
3
r
=
1.
1 + 0.5 sin θ
May 11, 2015
Obj: To write and analyze polar equations of conics and
write their equivalent rectangular equation.
Analyze the conic section given by the equation
below. Include values of e, a, b, and c. Determine the equivalent
Cartesian equation.
2.
r=
12
3 - 6 cos θ
May 11, 2015
Obj: To write and analyze polar equations of conics and
write their equivalent rectangular equation.
Find a polar equation and Cartesian equation for the conic that
has a focus at the pole and given polar coordinates as the
endpoints of its main axis.
( -6,
π
2
) and ( 2,
3π
2
)
May 11, 2015
HW:
Obj: To write and analyze polar equations of conics and
write their equivalent rectangular equation.
Day 1: (HR) (8.5) Pg. 618: 1-19odd
Day 2: (HR) (8.5) Pg.618: 33, 35, 37, 25