The locus of centre of the circle touching  x-axis and the line$y=x$  is

y=x makes angle ${45}^0$  with x-axis and angle bisector of this angle passes through the centre of the circle 

The angle bisector makes angle $22{\frac{1}{2}}^0$ with x-axis

Let $C\left(x_1,y_1\right)$  be the centre from figure,  $\tan 22{\frac{1}{2}}^0=\frac{y_1}{x_1}$

$\Rightarrow \left(\sqrt{2}-1\right)x_1=y_1$

$\therefore$ The locus of $\left(x_1,y_1\right)$  is $y=\left(\sqrt{2}-1\right)x$ .