The points $\left(x,1\right),\left(1,2\right),\left(0,y+1\right)$  are collinear, then $\frac{1}{x}+\frac{1}{y}=$

The slope of $AB=$  The slope of $BC$

$\frac{2-1}{1-x}=\frac{y+1-2}{0-1}$

$\frac{1}{1-x}=\frac{y-1}{-1}$

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

$-1=y-1-xy+x$

$y-xy+x=0$

$x+y=xy$

$\frac{x}{xy}+\frac{y}{xy}=1$

$\frac{1}{y}+\frac{1}{x}=1$