Question

What the is the polar form of `y = x^2y-x/y^2 +xy^2`?

Answer 1

`color(blue)[r^2=(-costheta)/[sin^3theta-r^2*sin^3theta*cos^2theta-r^2costheta*sin^4theta]]`

Explanation:

Note that

`color(red)[y=r*sintheta]`

`color(red)[x=r*costheta]`

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

`(r*sintheta)=(r*costheta)^2*(r*sintheta)-(r*costheta)/(r*sintheta)^2+(r*costheta)(r*sintheta)^2`

`rsintheta=r^3sintheta*cos^2theta-(costheta)/(rsin^2theta)+r^3costheta*sin^2theta`

`[rsintheta-r^3sintheta*cos^2theta-r^3costheta*sin^2theta]/1=-(costheta)/(rsin^2theta)`

`r^2sin^3theta-r^4*sin^3theta*cos^2theta-r^4costheta*sin^4theta=-costheta`

`color(blue)[r^2=(-costheta)/[sin^3theta-r^2*sin^3theta*cos^2theta-r^2costheta*sin^4theta]]`