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Q.
The differential equation representing the family of ellipses having foci either on the x-axis or on the y-axis, centre at the origin and passing through the point $(0, 3)$ is :
We know that general equation of ellipse is $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$
And passes through the point $(0,3)$
$\Rightarrow \frac{x^{2}}{a^{2}}+\frac{y^{2}}{9}=1$
Now differentiate the Eq. (1) with respect to $x$, we get
$\frac{2 x}{a^{2}}+\frac{2 y}{9} y^{\prime}=0 $
$\Rightarrow \frac{x}{a^{2}}=\frac{-y}{9} y'$
$ \Rightarrow \frac{1}{a^{2}}=\frac{-y}{9 x} y'$
From Eq. (1) and Eq. (2), differential equation is
$\frac{-x y}{9} y'+\frac{y'}{9}=1 $
$x y y'-y^{2}+9=0$