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Problems were equations in rectangular form are converted to polar form, using the relationship between polar and rectangular coordinates, are presented along with detailed solutions.
Problem 1: Convert the equation
2x 2 + 2y 2 - x + y = 0
to polar form.
Solution to Problem 1:
- Let us rewrite the equations as follows:
2 ( x 2 + y 2 ) - x + y = 0
- We now use the formulas giving the relationship between polar and rectangular coordinates: R 2 = x 2 + y 2, y = R sin t and x = R cos t:
2 ( R 2 ) - R cos t + R sin t = 0
- Factor out R
R ( 2 R - cos t + sin t ) = 0
- The above equation gives:
R = 0
or
2 R - cos t + sin t = 0
- The equation R = 0 is the pole. But the pole is included in the graph of the second equation 2 R - cos t + sin t = 0 (check that for t = Pi / 4 , R = 0). We therefore can keep only the second equation.
2 R - cos t + sin t = 0
or
R = (1 / 2)(cos t - sin t)
Problem 2: Convert the equation
x + y = 0
to polar form.
Solution to Problem 2:
- Use y = R sin t and x = R cos t into the given equation:
x + y = 0
R cos t + R sin t = 0
- Factor out R
R ( cos t + sin t ) = 0
- The above equation gives:
R = 0
or
cos t + sin t = 0
- The equation R = 0 is the pole. But the pole is included in the graph of the second equation cos t + sin t = 0 since this equation is independent of R. We therefore keep only the second equation.
cos t + sin t = 0
- The above equation may be written as.
tan t = - 1
- Solve for t to obtain
t = 3 pi / 4
- All points of the form (R , 3 pi / 4) are on the graph of the above equation. It is the equation of a line in polar form.
More references on polar coordinates and trigonometry topics.
Polar Coordinates.
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