Given:
2=(-x+2y)^2-y-x2=(−x+2y)2−y−x
x=rcosthetax=rcosθ
y=rsinthetay=rsinθ
Now,
2=(-rcostheta+2rsintheta)^2-rsintheta-rcostheta2=(−rcosθ+2rsinθ)2−rsinθ−rcosθ
2=r^2(-costheta+2sintheta)^2-r(sintheta+costheta)2=r2(−cosθ+2sinθ)2−r(sinθ+cosθ)
2=r^2((costheta)^2-2costheta(2sintheta)+(2sintheta)^2)-rsintheta-rcostheta2=r2((cosθ)2−2cosθ(2sinθ)+(2sinθ)2)−rsinθ−rcosθ
2=r^2cos^2theta-4r^2costhetasintheta+4r^2sin^2theta-rsintheta-rcostheta2=r2cos2θ−4r2cosθsinθ+4r2sin2θ−rsinθ−rcosθ
2=r^2(1+cos2theta)/2-4r^2(sin2theta)/2+4r^2(1-cos2theta)/2-r(sintheta+costheta)2=r21+cos2θ2−4r2sin2θ2+4r21−cos2θ2−r(sinθ+cosθ)
Simplifying further,
2=r^2(0.5+0.5cos2theta-2sin2theta+2-2cos2theta)-r(sintheta+costheta)2=r2(0.5+0.5cos2θ−2sin2θ+2−2cos2θ)−r(sinθ+cosθ)
2=r^2(2.5-1.5cos2theta-2sin2theta)-r(sintheta+costheta)2=r2(2.5−1.5cos2θ−2sin2θ)−r(sinθ+cosθ)
Or
r^2(2.5-1.5cos2theta-2sin2theta)-r(sintheta+costheta)-2=0r2(2.5−1.5cos2θ−2sin2θ)−r(sinθ+cosθ)−2=0
