Point of contact of $y=1-x$ w.r.t $y^2-y+x=0$ is

$y=1-x.....\left(1\right)$

$y^2-y+x=0.....\left(2\right)$

From 1 & 2 $y^2-y+1-y=0$

$\Rightarrow y^2-2y+1=0\Rightarrow (y-1{)}^2=0\Rightarrow y-1=0$

$\Rightarrow y=1\Rightarrow \left(1\right)\Rightarrow x=1-y\Rightarrow x=1-1\Rightarrow x=0$

$\therefore$point of contact $(0,1)$