The equation of the circle passing through (2, 1) and touching the coordinate axes is 

Let r be the radius of the circle.

Circle passes through (2, 1) $\Rightarrow$Circle lies in the first quadrant.

Circle touches the coordinate axes $\Rightarrow$Centre of the circle$=\left(r, r\right)$.

Equation of the circle is ${\left(x-r\right)}^2+{\left(y-r\right)}^2=r^2\Rightarrow x^2+y^2-2rx-2ry+r^2=0$

$\left(2, 1\right)$ lies on the circle $\Rightarrow 4+1-4r-2r+r^2=0\Rightarrow r^2-6r+5=0\Rightarrow \left(r-1\right)\left(r-5\right)=0$

$\Rightarrow r=1, 5.$

$\therefore$ The equation of the circles are   $x^2+y^2-2x-2y+1=0$ or $x^2+y^2-10x-10y+25=0$