The equation of the smallest circle passing through the points $(2, 2)$ and $(3, 3)$ is

If the smallest circle is drawn through the points $(2,2)$ and $(3,3)$ then, the point have to be the opposite end of the diameter of the circle.

Therefore, the centre of the circle is $C=(\dfrac{5}{2},\dfrac{5}{2})$.

The radius of the circle is $R=\dfrac{\sqrt{2}}{2}$ ....(using distance formula)

Hence equation of the circle is

$(x-\dfrac{5}{2})^{2}+(y-\dfrac{5}{2})^{2}=\dfrac{1}{2}$

$x^{2}+y^{2}-5x-5y+\dfrac{50}{4}=\dfrac{1}{2}$

Or

$x^{2}+y^{2}-5x-5y+\dfrac{25}{2}-\dfrac{1}{2}=0$

Or

$x^{2}+y^{2}-5x-5y+12=0$.