The equation of circle with centre (1, 2) and tangent $x + y - 5 = 0$ is

$\left|\frac{x+y-5}{\sqrt{2}}\right|\ =\ r\ \Rightarrow\frac{1+2-5}{\sqrt{2}}\ =\ r\ \therefore r\ =\ \sqrt{2}\$

$\left(x-1\right)^2+\left(y-2\right)²=2$

⇨${x^2} + {y^2} - 2x - 4y + 3 = 0$