$A B C D$ is a square with side 16 units let $a=16$ units
Centre of the circle $O\left(\frac{a}{2}, \frac{a}{2}\right)$


$$
A C=\sqrt{(a-0)^2+(a-0)^2}=\sqrt{2} a
$$
So, radius of circle $=\frac{A C}{2}=\frac{\sqrt{2} a}{2}=\frac{a}{\sqrt{2}}$
$\therefore$ Equation of circle
$$
\begin{array}{rlrl}
& \left(x-\frac{a}{2}\right)^2+\left(y-\frac{a}{2}\right)^2 & =\left(\frac{a}{\sqrt{2}}\right)^2 \\
\Rightarrow x^2+\frac{a^2}{4}-a x+y^2+\frac{a^2}{4}-a y & =\frac{a^2}{2} \\
\Rightarrow & x^2+y^2-a x-a y =0 \\
\Rightarrow & x^2+y^2 =a(x+y) \\
\text { here } & a =16 \\
\Rightarrow & x^2+y^2 =16(x+y) \\
\Rightarrow & x^2+y^2 =4.4(x+y) \\
\text { by comparing } & k =4
\end{array}
$$