Polar Coordinates
Advanced Level Pure Mathematics
Advanced Level Pure Mathematics
Calculus II
Introduction
2
Relations between Cartesian and Polar Coordinates
3
Sketch of Graphs in Polar Coordinates
5
Intersection of Two Curves in Polar Coordinates
10
Slope of a Tangent
11
Areas
13
Arc Lengths
15
Surfaces of Revolution
16
Prepared by K. F. Ngai
Page 1
Polar Coordinates
Advanced Level Pure Mathematics
Introduction
Cartesian Coordinate plane
y
Polar Coordinate plane
p (r,θ)
p (x, y)
vectorial angle θ
polor axis
(initial axis)
y
x
x
0 origin
(pole)
Example
y
P( 2 , 45)
p (1, 1)
2
1
x
0
45
1
N.B.
(1)
 is positive if it is measured anti-clockwise.
(2)
 is negative if it is measured clockwise.
P (4,
4
0

)
6
-

6

6

6
Q (4, - )
0
(3)
r can be negative.

3
A( 2, )
2
2

3

3
A( 2, )

3
2

3
0
B (- 2 , )
(4)
If r > 0 , P ( r,  ) = P ( r, 2n+ ) ,  n  Z .
e.g.
P ( r,  ) = P ( r, 3 ) = P ( r, - )
p (r, 3)
p (r,  )

r
p (r, -)
0
(5)
If r = 0 , O ( 0,  ) is called the pole or origin ,    R .
(6)
If r > 0 , Q ( -r,  ) = Q ( r, + ) .
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Polar Coordinates
Advanced Level Pure Mathematics

3
0
4


)
3
3
Q is an imaginary point (which r<0) and P is a real point (which r>0) .
Q (4, + ) or Q (-4,
(7)
Relations between Cartesian and Polar Coordinates
r  x 2  y 2
x

r
cos



 y  r sin   
1 y


tan


x
y
P (x, y)  P (r, )
r
0
Example
(a)
r
y

x
Express the Cartesian coordinates in polar coordinates
(a)
(b)
( 2 3 , 2) ,
( 2, 
x (polar axis)
2)
x 2  y 2  12  4  4
2

1
1 1
  tan
 tan
2 3

3
6

so the polar coordinates= ( 4, )
6
(b)
Example
Express the polar coordinates in Cartesian coordinates
Prepared by K. F. Ngai
Page 3
Polar Coordinates
(a)
(-2 , ) ,
(b)
(2  3 ,

6
Advanced Level Pure Mathematics
)
Definition
Polar Equations : r = f()
Example
r=2cos , r=2(1-sin) , r=3 , etc.
Example
Investigate the loci represented by the following polar equations.
(a)
=
(a)
 =  , where  is a constant.
(b)
r = a , where a is a constant.
 tan 1

y

x
y
 tan   m (say)
x
 y  mx
it is a straight line with inclination  to the initial line and passing through the pole.
(b)
Example
Express the following loci in polar coordinates :
Prepared by K. F. Ngai
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Polar Coordinates
Advanced Level Pure Mathematics
(a)
The straight line with normal form x cos   y sin   p .
(b)
2
2
The circle x  y  2ax  0 .
Sketch of Graphs in Polar Coordinates
Plot the polar curve r  2 cos for 0    2 .
Example
Solution

0
r
2

4
2

3

2
1
0
3
4
- 2

….
3
2
-2 ….
….
2
0 ….
2
y
P( r,  )
r
0

x (polar axis)
1
N.B.
(1)
2
r=2cos
r  2 cos   x 2  y 2  2
x
x  y2
2
 ( x  1) 2  ( y  0) 2  12
(2)
(which is a circle)
If f (  )  f ( ) , the polar curve is symmetric with respect to the initial line.
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Polar Coordinates
Advanced Level Pure Mathematics
(r, )

0
e.g.
r=2cos , r=2(1-cos)
-
(r ,- )
If f ( )   f ( ) , or f ( )  f (   ) , the curve is symmetric with respect to the
(3)
vertical line  

.
2
(-r, -)=(r, -)
(r, )
e.g.
r=2sin , r=a(1+sin)
 -
0
If f ( )  f (   ) , or   g (r )  g (r ) , the curve is symmetric with respect to the pole.
(4)
(r, )
0
r2 = a2 sin2
e.g.
(-r, )
=(r, +)
I.
Cardioids
Sketch the graph of the polar equation r  2(1  cos ) for 0    2 .
Example
Solution
Let f() = 2(1-cos)
f() = 2[1-cos(-)] = f()
 The graph is symmetric wiyh respect to the initial line.

6

0
r
0 2- 3

4
2- 2

3

2
2
3
1
2
3
3
4
5
6
2+ 2 2+ 3

4
The graph is
R=2(1-cos)
(4, )
Example
0
Sketch the graph of the polar equation r  2(1  cos ) for 0    2 .
Prepared by K. F. Ngai
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Polar Coordinates
Advanced Level Pure Mathematics
Since f(-)=2[1+cos(-)]=2(1+cos)=f() so the graph is symmetric w.r.t. the initial line.
N.B.
r=a(1+sin)
r=a(1-sin)

2
(2a, )
(a,)
0
(0, )
0
(a, 0)
(2a ,
II.
(a, 0)
3
)
2
Limacons
Example
Sketch the graph of the polar equation r  2  4 cos .
N.B. r=a+bcos (0<a<b)
r=a+bcos (0<b<a)
Prepared by K. F. Ngai
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Polar Coordinates
Advanced Level Pure Mathematics
0
0
r=a+bsin (0<b<a)
r=a+bsin (0<a<b)
0
0
III.
n-Leafed Roses
r=acos3 (a>0 , 0    2 )
r=asin3 (a>0 , 0    2 )
=
=

6
=
5
6
r=acos2 (a>0 , 0    2 )
2
3
=

3
r=asin2 (a>0 , 0    2 )
Example
Sketch the graph of the polar equation r  a sin 3 , where a>0.
Example
Sketch the graph of the polar equation r  a cos 2 , where a>0.
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Polar Coordinates
Advanced Level Pure Mathematics
IV.
Two-Leafed Lemniscates
r2=a2cos2 (a>0 , 0    2 )
V.
r2=a2sin2 (a>0 ,
0    2 )
Spirals of Archimedes.
r=a (a>0 ,   0)
r  e a (a>0 ,   0)
r  e a (a<0 ,   0)
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Polar Coordinates
Advanced Level Pure Mathematics
Intersection of Two Curves in Polar Coordinates
Example
Find the points of intersection of the circle r  2 and the cardioid r  2(1  cos ) .
Example
Find the points of intersection of the circle r  2 cos and the four-leafed rose r  sin 2 .
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Polar Coordinates
Advanced Level Pure Mathematics
Slope of a Tangent
dr
 dx

cos

 r sin
 d
 x  r cos
d

 y  r sin   dy
dr

  sin
 r cos
d
 d
dr
sin 
 r cos
dy
d



dr
dx
cos
 r sin 
d
dr
tan 
r
d


dr
 r tan 
d
The angle  between the tangent at P to the polar
curve r=f() and radius vector OP is given by
y
tan  
r=f()
r
dr
d
tangent at point p

p(r, )

x
0
Example
Given the polar curve r  2  2 cos , 0    2 .
Prepared by K. F. Ngai
Page 11
Polar Coordinates
(a)
is

4
Find
(i)
(ii)
(iii)
(iv)
the slope of tangent at  

Advanced Level Pure Mathematics
,
4
the points at which the tangent is horizontal,
the points at which the tangent is vertical,
the values of  at which the angle between the radius vector and tangent
.
(b)
Sketch the polar curve.
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Polar Coordinates
Advanced Level Pure Mathematics
Areas
Theorem
The area enclosed by the polar curve r  f ( ) betweem the lines = and = is given by
r=f()

1 2
r d
 2
A
 =
=
where r  f ( )  0 on [ , ].
0
Example
Find the area enclosed by the cardioid r  a(1  cos ) , a>0.
Example
Find the area of the inner loop of the curve r  2  4 cos .
Solution
2 4 
Since the inner loop bounded by the curve corresponds to the interval 
.
,
3 
 3
N.B.
(1) r  f ( ), r  g ( )
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Polar Coordinates
Advanced Level Pure Mathematics
Area enclosed
=
 1
1
 2  f ( )  g ( ) d =  2  f ( )
2
2
2
 g ( ) d
2
=
=
r = f()
r = g()
(2) If the pole O lies within a closed polar curve, then
the area enclosed=
Example
(a)
(b)

2
0
1 2
r d .
2
Find the area inside the circle r  5 cos and outside the limacon r  2  cos .
Find the area inside the circle r  5 cos and the limacon r  2  cos .
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Polar Coordinates
Advanced Level Pure Mathematics
Arc Lengths
The arc length S of the curve r  f ( ) form= to = is

S
=  ds

= 
dy
ds
dx 2  dy 2
2
dx
2

 dx   dy 

 
 d
 d   d 

dr
dr




 r sin     sin 
 r cos   d
 cos 
d
d





 dr 
2

  r d
 d 
= 
= 
= 
r=f()
2
2
2
provided that r is differentiable on ( , ) and the curve does not intersect itself on ( , ).
Example
Find the length of the circumference of the cardioid r  a(1  cos ) , where a>0.
Example
Find the length of the curve of the circle r  5 cos that is outside the limacon r  2  cos .
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Polar Coordinates
Advanced Level Pure Mathematics
Surfaces of Revolution
Theorem
The area of surface revolution S of the curve from = to = about the inital line is

S   2yds

2

 dr 
2
  2r sin  
  r d

 d 
r=f()

ds
y

0
Theorem
The area of surface revolution S of the curve from = to = about the line  

2
(i.e. y-axis)
is

S   2xds

2
 dr 
2
  2r cos  
  r d

 d 


x
ds

0
Example
Find the surface area generated by revolving the circle r  2a sin  , a>0 about the line  

2
.
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Polar Coordinates
Advanced Level Pure Mathematics
Example
Find the surface area generated by revolving the cardioid r  a(1  cos ) , where a>0 , about
the initial line.
Example
The equation of a curve C in polar coordinates is
r  1 sin  , 0    2 .
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Page 17
Polar Coordinates
Advanced Level Pure Mathematics
(a)
Sketch curve C.
(b)
Find the area bounded by curve C.
[HKAL94] (5 marks)
Example
y
p
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Polar Coordinates
Advanced Level Pure Mathematics
Q

0
a
2a
x
Let C be the circle given by the polar equation r=2acos (where a>0) , P be a variable point on C and O
be the origin. Let Q be a point lying on the line through O and P such that P and Q are on the same side
of O and
OP  OQ  a 2 .
Show that the Cartesian equation of the locus of Q is x 
a
.
2
[HKAL93] (5 marks)
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Page 19