While going through an exercise of surface integration, I got confused in this problem.The surface is the intersection of sphere $S:x^2+y^2+z^2-1=0$ and the plane $P:y-x=0$. Clearly, the curve of intersection is the circle $S=0,P=0$ whose projection on $XZ$ or $YZ$ plane is an ellipse. It seems that it can be obtained by substituting $y=x$ in $S=0$ to get $2x^2+z^2=1$ (point of confusion) but I don't think this is the projection as the calculations doesn't support the geometry.
Kindly correct me what is wrong in my approach. Thanks