Question

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 :

• A. $xy{y}^{\prime \prime }+x(y^{\prime }{)}^2-y{y}^{\prime }=0$
• B. $x+y{y}^{\prime \prime }=0$
• C. $xy{y}^{\prime }+y^2-9=0$
• D. $xy{y}^{\prime }-y^2+9=0$

Answer 1

Answer: (D)

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
$\frac{2x}{a^2}+\frac{2y}{9}{y}^{\prime }=0$
$\Rightarrow \frac{x}{a^2}=\frac{-y}{9}{y}^{\prime }$
$\Rightarrow \frac{1}{a^2}=\frac{-y}{9x}{y}^{\prime }$
From Eq. (1) and Eq. (2), differential equation is
$\frac{-xy}{9}{y}^{\prime }+\frac{y^{\prime }}{9}=1$
$xy{y}^{\prime }-y^2+9=0$