Question

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

Answer 1

By substitution

Explanation:

Use the substitutions

`x= rcos(theta)" "` and `" "y= rsin(theta)`

Then

`rsin(theta) = 3r^2cos^2(theta) - 2rcos(theta) - rcos(theta)r^2sin^2(theta)`

`r^3cos(theta)sin^2(theta) -3r^2cos^2(theta)+2rcos(theta) +rsin(theta) = 0`

We know `sin^2(theta) + cos^2(theta) = 1`

`:. sin^2(theta) =1- cos^2(theta)`

Substituting into the equation above gives

`r^3cos(theta)(1-cos^2(theta))- 3r^2cos^2(theta)+2rcos(theta) +rsin(theta) = 0`

`r^3cos(theta) -r^3cos^3(theta) -3r^2cos^2(theta) +2rcos(theta) +rsin(theta) = 0`

`r^3cos^3(theta)+3r^2cos^2(theta)-(r^3+2r)cos(theta) -rsin(theta)=0`