Translate the equation x + y = 2x^{2} into polar coordinates.

Replace {x} with {rcosθ} and replace {y} with {rsin θ}.

x + y = 2 x^{2} rcos θ + r sin θ = 2 (rcos θ)^{2} rcos θ + rsinθ& = 2r^{2}cos^{2}θ

One solution to this equation is r = 0. If r isn't zero, we can divide both sides of the equation by r to find cosθ + sinθ = 2rcos^{2}θ. Since polar equations are usually written in the form r = f(θ) we rearrange this a bit more to find r = (cosθ + sinθ)/2cos^{2}θ.