Answer
Center: $(1,2)$
Radius = $\sqrt 2$ units
Equation: $(x-1)^2 +(y-2)^2=2$
Work Step by Step
The center $(h,k)$ will be the midpoint of the endpoints of the diameter:
$$(h,k)=\left(\frac{2+0}{2}, \frac{1+3}{2}\right)=(1,2)$$
The raduis r$ $will be the half of the diameter
The diameter $d$ is equal to the distance between its endpoints $(0, 1)$ and $(2, 3)$.
Thus,
$$d=\sqrt{(2-0)^2+(3-1)^2}=\sqrt8=\sqrt{4(2)}=2\sqrt2$$
Hence, the radius is:
$$r=\dfrac{2\sqrt2}{2}=\sqrt2$$
Therefore, the standard form of the equation is
$$(x-1)^2+(y-2)^2=(\sqrt2)^2\\
(x-1)^2+(y-2)^2=2$$
