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(A) The equation of a family of circles of radius $r$ passing through the origin and having its centre on the $y$-axis is given by $(x - 0)^2 + (y - r

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$1$

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(A) The equation of a family of circles of radius $r$ passing through the origin and having its centre on the $y$-axis is given by $(x - 0)^2 + (y - r)^2 = r^2$.Expanding this,we get $x^2 + y^2 - 2ry + r^2 = r^2$,which simplifies to $x^2 + y^2 - 2ry = 0$.This equation contains only one arbitrary constant,$r$.The order of a differential equation is equal to the number of independent arbitrary constants in the general solution of the corresponding family of curves.Since there is only one arbitrary constant $r$,the order of the differential equation is $1$.

The order of the differential equation of all circles of radius $r$,having centre on the $y$-axis and passing through the origin is:

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