Equation of circle passing through origin & having centre at the line $$y=x$$ is

$$\begin{aligned} & (x-t)^2+(y-t)^2=2 t^2 \\ & x^2+y^2+t^2+t^2-2 t x-2 t y=2 t^2 \\ & x^2+y^2=2 t(x+y) \end{aligned}$$

Now differentiate

$$\begin{aligned} & 2 x+2 y y^{\prime}=2 t\left(1+y^{\prime}\right) \\ & t=\frac{x+y y^{\prime}}{1+y^{\prime}} \end{aligned}$$

Now, $$x^2+y^2=2\left(\frac{x+y y^{\prime}}{1+y^{\prime}}\right)(x+y)$$

$$\begin{aligned} & x^2+y^2+x^2 \frac{d y}{d x}+y^2 \frac{d y}{d x} \\ & =2\left(x^2+x y+x y \frac{d y}{d x}+y^2 \frac{d y}{d x}\right) \\ & d x\left(x^2+y^2-2 x^2-2 x y\right)=d y\left(2 x y+2 y^2-x^2-y^2\right) \\ & d x\left(x^2-y^2+2 x y\right)=d y\left(x^2-y^2-2 x y\right) \end{aligned}$$