If the line y=x touches the curve

Given that line $y=x$ touches the curve $y=x^{2}+b x+c$ at a point $(1,1)$

Slope of line $=1$

Slope of tangent to the curve $=1$

$\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=2 \mathrm{x}+\mathrm{b}$

$\Rightarrow 2 \mathrm{x}+\mathrm{b}=1$

$\Rightarrow 2+\mathrm{b}=1$

$\Rightarrow \mathrm{b}=-1$

Putting this and $x=1$ and $y=1$ in the equation of the curve,

$1=1-1+c$

$\Rightarrow c=1$