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ALGEBRA
  Polar coordinate system
  The polar coordinate system is a two-dimensional 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 x-axis, 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, -3) 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 y-axis
A vertical line, x = c is represented by the equation
r cosq = c   or
Lines parallel to the x-axis
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 x-axis
Thus, the equation of a line through the origin is represented by the equation  q = a  in polar coordinates.
Polar equation of a line
As
the polar equation of a line

where p is the distance of the line from the pole O and j is the angle that the segment p makes with the polar axis.

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
Therefore,
 :: 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

(r0, j) and radius R.

Using the law of cosine,
r2 + r02 - 2rr0 cos(q - j) = R2
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,   r2 = x2 + y2   and   x2 + y2 = R2   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    r2 = - 4r cosq,
and by using polar to Cartesian conversion formulas,   r2 = x2 + y2   and   x = r cosq
obtained is         x2 + y2 = - 4x
                 x2 + 4x + y2 = 0
        or      (x + 2)2 + y2 = 4
the equation of a circle with radius R = 2 and the
center at (-2, 0).
 
 
 
 
 
 
 
 
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