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

y=0,y=x are tangents to the circle from (0,0)

Locus of centre is angle  bisector of the angle between the lines y = 0, y = x.

$$\begin{gathered} \Rightarrow Locus of centre is y=\left(\tan \frac{\pi }{8}\right)x \\ \therefore y=\left(\sqrt{2}-1\right)x \end{gathered}$$