Equation of the circle passing through $A(1,2),B(5,2)$ so that the angle subtended by $AB$  at points on the circle is $\mathrm{\pi }/4$ is

The angle subtended by $\overline{AB}$ at a point on the circle is $\mathrm{\pi }/4$ 

the angle subtended by $\overline{AB}$ at the circle $C$  of the circle is  $2\left(\frac{\pi }{4}\right)=\frac{\pi }{2}$

Let ‘r’ be the radius of the circle $AB=4.$

$C{A}^2+C{B}^2=A{B}^2\Rightarrow r^2+r^2=4^2\Rightarrow 2r^2=16\Rightarrow r=2\sqrt{2}$

Midpoint of $\overline{AB}\text{ is }M(3,2)$