For converting a equation with Cartesian coordinates into a polar equation, one can use the relations x=rcosthetax=rcosθ and y=rsinthetay=rsinθ
Hence, y=(3x-2y)^2-x^2y-5y^2y=(3x−2y)2−x2y−5y2 can be written as
rsintheta=(3rcostheta-2rsintheta)^2-(rcostheta)^2(rsintheta)-5(rsintheta)^2rsinθ=(3rcosθ−2rsinθ)2−(rcosθ)2(rsinθ)−5(rsinθ)2 or
rsintheta=9r^2cos^2theta+4r^2sin^2theta-12r^2sinthetacostheta-r^3cos^2thetasintheta-5r^2sin^2thetarsinθ=9r2cos2θ+4r2sin2θ−12r2sinθcosθ−r3cos2θsinθ−5r2sin2θ
or rsintheta=9r^2(1-sin^2theta)-r^2sin^2theta-6r^2sin2theta-r^3cos^2thetasinthetarsinθ=9r2(1−sin2θ)−r2sin2θ−6r2sin2θ−r3cos2θsinθ
or sintheta=9r-10rsin^2theta-6rsin2theta-r^2cos^2thetasinthetasinθ=9r−10rsin2θ−6rsin2θ−r2cos2θsinθ
or r^2cos^2thetasintheta+r(6sin2theta+10sin^2theta-9)+sintheta=0r2cos2θsinθ+r(6sin2θ+10sin2θ−9)+sinθ=0
