dxdy=2xyy2−x2,x∈(0,∞)
put y=vx
xdxdv+v=2vv2−1
v2+12vdv=−xdx
Integrate,
ln(v2+1)=−lnx+C
ln(x2y2+1)=−lnx+C
put x=1,y=1,C=ln2
ln(x2y2+1)=−lnx+ln2
⇒x2+y2−2x=0 (Curve C1)
Similarly,
dxdy=x2−y22xy
Put y=vx
x2+y2−2y=0

Required area =2∫01(2x−x2−x)dx=2π−1 sq. units