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# Form the differential equation of the family of circles having centre on $y$-axis and radius $3$ units.

$\begin{array}{1 1} (A)\;(x^2 - 9)(y')^2 + x^2 = 0 \\ (B)\;(x^2 - 9)(y')^2 - x^2 = 0 \\ (C)\;(x^2 - 9)(y')^2 + x^3 = 0 \\ (D)\;(x^2 - 9)(y')^2 + x^4 = 0 \end{array}$

Toolbox:
• Equation of a circle having centre on the y axis and radius a units is $x^2 + (y-h)^2 = a^2$
Step 1:
From the information in the tool box we obtain the equation of the circle as
$x^2 + (y-h)^2 = 9$-----(1)
On differentiating on both sides with respect to $x$ we get,
$2x + 2(y-h).y' = 0$
$y-h =\large\frac{ x}{y'}$
Step 2:
Substituting for $y-h$ in equ(1) we get
$x^2 + (\large\frac{x}{y'})^2$$= 9$
on simplifying and rearranging we get
$(x^2 - 9)(y')^2 + x^2 = 0$
This is the required differential equation.