The differential equation of the family of circles with fixed radius r and with centre on y-axis, is

The equation of the family of circles is 

$x^2+(y-k{)}^2=r^2$

where k is a parameter.

Differentiating with respect to x, we get

$2x+2(y-k)y_1=0\Rightarrow y-k=-\frac{x}{y_1}$

Eliminating k from (i) and (ii), we obtain

$x^2+\frac{x^2}{y_1^2}=r^2\Rightarrow x^2=\frac{r^2{y}_1^2}{1+y_1^2}\Rightarrow x^2\left(y_1^2+1\right)=r^2{y_1}^2$