Question

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

Answer 1

`0=r^2-r^2sin(2theta)-1/2 r^3costhetasin(2theta)+1/2r^3sinthetasin(2theta)-rsintheta`

Explanation:

`0=x^2-2xy+y^2-x^2y+xy^2-y`

`0=x^2+y^2-2xy-x^2y+xy^2-y`

use formulas:
`x^2+y^2=r^2,x=rcostheta,y=rsintheta`

`0=r^2-2rcosthetaxxrsintheta-r^2cos^2thetaxxrsintheta+rcos thetaxxr^2sin^2theta-rsintheta`

`0=r^2-2r^2sinthetacostheta-r^3sinthetacos^2theta+r^3sin^2thetacostheta-rsintheta`

`0=r^2-r^2(2sinthetacostheta)-1/2 r^3costheta(2sinthetacostheta)+1/2r^3 sintheta(2sinthetacostheta)-rsintheta`

`0=r^2-r^2sin(2theta)-1/2 r^3costhetasin(2theta)+1/2r^3sinthetasin(2theta)-rsintheta`