MTH 253Calculus (Other Topics)

Chapter 10 – Conic Sections and Polar Coordinates

Section 10.8 – Conic Sections in Polar Coordinates

Copyright © 2009 by Ron Wallace, all rights reserved.

Conics - Reminder

The cross product term (Bxy) can be removed by rotation of axis where

2 2 0Ax Bxy Cy Dx Ey F

112

0

tan if

45 if

B A CA C

A C

' cos 'sinx x y

' sin 'cosy x y

Conics - Reminder

Nothing: AC > 0 & D2/(4A) + E2/(4C) – F < 0 C = E = 0 & D2 – 4AF < 0 A = D = 0 & E2 – 4CF < 0

Point: AC > 0 & F = D2/(4A) + E2/(4C)

Line(s): A = 0, C = 0, & D and/or E ≠ 0 C = E = 0 & D2 – 4AF ≥ 0 A = D = 0 & E2 – 4CF ≥ 0 A > 0, C < 0, & F = D2/(4A) – E2/(4C) A < 0, C > 0, & F = –D2/(4A) + E2/(4C)

2 2 0Ax Cy Dx Ey F

NOTE: These are known as the “degenerate” cases.

Conics - Reminder

Parabola: A = 0 & C ≠ 0 A ≠ 0 & C = 0

Circle: A = C ≠ 0

Ellipse: AC > 0 & A ≠ C

Hyperbola: AC < 0

2 2 0Ax Cy Dx Ey F

NOTE: These assume non-degenerate cases.

Conics in Polar Coordinates Some applications that use conics,

especially astronomy, work better with polar coordinates.

Lines in Polar Coordinates Lines through the pole (i.e. origin)

0 0

Lines in Polar Coordinates Lines NOT through the pole

00cos r

r

0 0,r

,r

0

Using the right triangle …

NOTE: The blue line is perpendicular to the red line.

0 0cosr r

Converting Polar Lines to Cartesian Lines

0 0cosr r

0 0 0cos cos sin sinr r r

0 0 0cos sinx y r

Converting Cartesian Lines to Polar Lines

y mx b

1perpendicular my x

1. Find the point of intersection of these two lines.

2. Convert that point into polar coordinates: (r0,0)

3. Give the polar equation … 0 0cosr r

Circles in Polar Coordinates

0 0,r

,r

0

a

Using the law of cosines …

2 2 20 0 02 cosa r r r r

If the circle passes through the pole, r0 = a …

02 cosr a

Circles in Polar Coordinates

0 0,r

,r

0

a

Circles through the pole with the center on the x-axis.

2 cosr a

Circles through the pole with the center on the y-axis.

2 sinr a

Reminder: The Focus-Directrix Property of Conics

Given a point F (focus) a line not containing F (directrix) a constant e (eccentricity)

A conic is the set of all points P where PF = e · PD

e=1 parabola 0<e<1 ellipse e>1 hyperbola

F

P

D

Polar Equations of Conics For polar equations of conics

focus at the pole (i.e. origin) directrix a vertical line: x = k > 0

PF = r PD = k – rcos Therefore, since PF = e · PD …

r = e(k - rcos) Solving for r …

PF = e · PD

e=1 parabola 0<e<1 ellipse e>1 hyperbola

F

P D

k

(r,)

1 or

1 cos cose

ek kr re

Examples … Describe the graphs of the following equations

(type of conic, directrix, intercepts, vertices)

PF = e · PD

e=1 parabola 0<e<1 ellipse e>1 hyperbola

F

P D

k

(r,)

51 cos

r

1 or

1 cos cose

ek kr re

203 4cos

r

103 2cos

r

Polar Equations of Conics Other orientations …

Directrix: x = –k

Directrix: y = k

Directrix: y = –k

1 or

1 cos cose

ek kr re

1 or

1 sin sine

ek kr re

1 or

1 sin sine

ek kr re

More Examples Describe the graphs of the following equations

(type of conic, directrix, intercepts, vertices)

74 4sin

r

67 4sin

r

121 4sin

r

203 4cos

r