

ALGEBRA


Polar coordinate system

The
polar coordinate system is a twodimensional coordinate system
in which each point P
on a plane is determined by the length of its position vector r
and the angle q
between it and the positive direction of the xaxis,
where 0 <
r
< + oo
and 0
<
q
< 2p.


Polar
and Cartesian coordinates relations


Since the inverse tangent function
(arctan or tan^{}^{1})
returns values in the range p/2
< q <
p/2, then 
for points lying in the 2nd or 3rd quadrant 
and for points lying in the 4th quadrant 




Example:
Convert Cartesian coordinates (1,
Ö3)
to polar coordinates.

Solution: 

and since the point lies in the 3rd quadrant,
then 










::
Equation
of a line in polar form

Lines
parallel to the axes, horizontal and vertical lines

Lines
parallel to the yaxis 
A
vertical line, x
=
c is
represented by the equation 
r
cosq
=
c
or 




Lines
parallel to the xaxis 
A
horizontal line, y
=
c is
represented by the equation 
r
sinq
=
c
or 




Lines
running through the origin or pole (radial lines)

The
equation of a line through the origin or pole that makes an angle
a with the positive
xaxis 

Thus,
the
equation of a line through the origin is
represented by the equation q
=
a
in polar
coordinates. 

Polar
equation
of a line


Example:
Write polar equation of the
line passing through points (4,
0)
and (0, 4).

Solution: Using polar equation of a line 

Proof,
using Cartesian to polar
conversion formulas. 
The
intercept form of the line 

or 


x
+ y =
4,
y
=
x
+ 4 





::
Equation
of a circle in polar form

General
equation of a circle in polar coordinates 
The
general equation of a circle with a center at 
(r_{0},
j)
and radius R. 
Using
the law of cosine, 
r^{2}
+ r_{0}^{2}

2rr_{0}
cos(q

j)
=
R^{2} 



Polar
equation of a circle with a center on the polar axis running
through the pole

Polar
equation of a circle with radius R
and a center on the polar axis running through the pole O
(origin). 
Since 

then, 
r
=
2R
cosq 



Polar
equation of a circle with a center at the pole

Since,
r^{2}
=
x^{2}
+ y^{2}
and x^{2}
+ y^{2}
=
R^{2}
then r
=
R 
is
polar
equation of a circle with radius R
and a center at the pole (origin). 


Example:
Convert the polar
equation of a circle r
=

4
cosq
into Cartesian coordinates.

Solution:
As, r
=

4
cosq
then r^{2}
=

4r
cosq, 
and
by using polar to Cartesian conversion formulas, r^{2}
=
x^{2}
+ y^{2}
and x
=
r cosq 
obtained
is
x^{2}
+ y^{2}
=

4x 
x^{2}
+ 4x
+ y^{2}
= 0

or
(x
+ 2)^{2} + y^{2}
= 4

the
equation of a circle with radius R
= 2 and
the

center
at (2,
0).




















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