Post on 18-Jan-2018
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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