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

Answer 1

The correct option is A $xy{y}^{\prime }–y^2+9=0$

According to the question general equation of ellipse is
$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\text{As the ellipse passes through}(0,3)$
$\frac{x^2}{a^2}+\frac{y^2}{9}=\mathrm{1........}\left(1\right)$
Now differentiate w.r.t $x$
$\frac{x}{a^2}=-\frac{y}{9}\frac{dy}{dx}\Rightarrow \frac{1}{a^2}=-\frac{y}{9x}{y}^{\prime }........\left(2\right)$
From (1) and (2) Differential equation is
$xy{y}^{\prime }-y^2+9=0$