The equation of family of circles of fixed radius ' $r$ ' with centres on the $y$ -axis is
$(x-0)^{2}+(y-a)^{2}=r^{2}\,\,\,\,\,\,\,\dots(i)$
On differentiating w.r.t. $x$, we get
$2 x+2(y-a) \frac{d y}{d x}=0 $
$\Rightarrow \, \frac{d y}{d x}=-\frac{x}{y-a} $
$\Rightarrow \, (y-a)=\frac{-x}{(d y / d x)}$
On putting this value in Eq. (i), we get
$x^{2}+\frac{x^{2}}{\left(\frac{d y}{d x}\right)^{2}}=r^{2}$
$\Rightarrow \, x^{2}\left\{1+\left(\frac{d y}{d x}\right)^{2}\right\}$
$=r^{2}\left(\frac{d y}{d x}\right)^{2}$
Hence, order $\to$ highest order derivative $=1$
and degree $\to$ power of highest order derivative $=2$