How do you convert #3y= 2x^2-2xy-7y^2 # into a polar equation?

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Answer 1 · 2017-03-30T18:13:54

Substitute #rcos(theta)# for #x# and #rsin(theta)# for y.
Write #r# as a function of #theta#

Given: #3y= 2x^2-2xy-7y^2#

Here is the graph of the Cartesian equation:

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Substitute #rcos(theta)# for #x# and #rsin(theta)# for y.

#3rsin(theta)= 2(rcos(theta))^2-2(rcos(theta))(rsin(theta))-7(rsin(theta))^2

Write #r# as a function of #theta#

#3sin(theta)r = (2cos^2(theta) -2cos(theta)sin(theta)-7sin^2(theta))r^2#

Please observe that we can safely divide by r, because that will only eliminate the trivial root #r = 0#:

#3sin(theta) = (2cos^2(theta) -2cos(theta)sin(theta)-7sin^2(theta))r#

Divide by the coefficient in front of r:

#r = (3sin(theta))/(2cos^2(theta) -2cos(theta)sin(theta)-7sin^2(theta))#

Here is the graph the polar equation.

Desmos.com

This proves that the conversion is done properly.