Let centre of circle on X-axis be $(h, 0)$
The radius of circle will be h
$\therefore $ The equation of circle having centre $(h, 0)$ and radius $h$ is
$(x-h)^{2} + (y-0)^{2} = h^{2}$
$\Rightarrow x^{2}+h^{2}-2hx+y^{2} = h^{2}$
$\Rightarrow x^{2}-2hx+y^{2}=0 \, \dots(i)$
On differentiating both sides w.r.t x, we get
$2x-2h+2y \frac{dy}{dx}=0$
$\Rightarrow h=x+y \frac{dy}{dx}$
On putting $h = x+y \frac{dy}{dx}$ in Eq. (i), we get
$x^{2}-2\left(x+y \frac{dy}{dx}\right) x+y^{2}=0$
$\Rightarrow -x^{2}+y^{2}-2xy \frac{dy}{dx}=0$
$\Rightarrow \left(x^{2}-y^{2}\right)+2xy \frac{dy}{dx}=0$
