Let mid point of the chord $A B$ is $C\left(x_{1}, y_{1}\right)$.
In $\Delta C O B, \sin \frac{\pi}{4}=\frac{B C}{O B}$
$\Rightarrow \,\,\,\,\,\, \frac{1}{\sqrt{2}}=\frac{B C}{2}$
$\Rightarrow \,\,\,\,\,\, B C=\sqrt{2}$
Using Pythagoras theorem
$O B^{2}=O C^{2}+C B^{2}$
$\Rightarrow \,\,\,\,\,\,(2)^{2}=x_{1}^{2}+y_{1}^{2}+(\sqrt{2})^{2} $
$\Rightarrow \,\,\,\,\,\, x_{1}^{2}+y_{1}^{2}=2$
Hence, locus of mid point of chord is
$x^{2}+y^{2}=2$
