change x^4-xy-2y^4=3y to cartesian?

Question

change x^4 - xy - 2y^4 = 3y from polar to cartesian (rectangular)

I tried working it out but it doesn't look right. Am I suppose to solve for r? or can you leave it as something else? I am a little confused.

I got
(rcos(θ) )^4/rsin(θ) - r cos(θ) - 2(rsin(θ))^3 = 3
I don't know what to do after that.

Help would be appreciated. Thanks

Sorry I mean change to polar from cartesian. Sorry about that. I was looking at another problem

Answer 1

Agreed, you mean cartesian to polar.
What you have is correct (as long as y ≠ 0), but I think you'd do better not to divide through by y. Then you get
(r cos θ)^4 - r^2 cos θ sin θ - 2(r sin θ)^4 = 3 r sin θ
<=> r^4 cos^4 θ - r^2 cos θ sin θ - 2r^4 sin^4 θ - 3r sin θ = 0
Note that cos^4 θ - sin^4 θ = (cos^2 θ - sin^2 θ) (cos^2 θ + sin^2 θ) = cos^2 θ - sin^2 θ = cos 2θ, so we can simplify to
r^4 (cos 2θ - sin^4 θ) - r^2 sin θ cos θ - 3r sin θ = 0.
It's a matter of choice as to whether you prefer sin θ cos θ to (1/2) sin 2θ; you can look for other simplifications, but I don't think there's much more you can do from this point.

Answer 2

You meant to say change the problem from cartesian to polar, and yea your answer seems right. Just simplify.

Answer 3

I believe you want to go from Cartesian to polar.

Use these substitutions:

R=Sqrt(x2+y2);
Theta=ArcTan(Y/X)

Good luck!