Question

The differential equation of the family of ellipse having foci on $Y$-axis and centre at origin is

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

Answer 1

Answer: (C)

The equation of family of ellipse is of the form
$\frac{x^2}{b^2}+\frac{y^2}{a^2}=\mathrm{1....}$(i)
(since, foci is on $Y$-axis, so we draw a vertical ellipse)
On differentiating Eq. (i) w.r.t. x, we get
$\frac{d}{dx}\left(\frac{x^2}{b^2}\right)+\frac{d}{dx}\left(\frac{y^2}{a^2}\right)=\frac{d}{dx}(1).....$(ii)
$\Rightarrow \frac{1}{b^2}2x+\frac{1}{a^2}2y{y}^{\prime }=0$
$\frac{y{y}^{\prime }}{x}=-\frac{a^2}{b^2}$
Again, differentiating w.r.t. $x$, we get
$\Rightarrow \frac{x\frac{d}{dx}\left(y{y}^{\prime }\right)-y{y}^{\prime }\frac{d}{dx}(x)}{x^2}=0$
$[$ using uqotient rule $\frac{d}{dx}\left(\frac{u}{v}\right)=\frac{v\frac{d}{dx}(u)-u\frac{d}{dx}(v)}{v^2}]$
$\Rightarrow \frac{x\left[y{y}^{\prime \prime }+{\left(y^{\prime }\right)}^2\right]-y{y}^{\prime }\cdot 1}{x^2}=0$
$[$ using product rule $\frac{d}{dx}(u\cdot v)=\left(u\frac{d}{dx}v+v\frac{d}{dx}u\right)]$

$\Rightarrow x{\left(y^{\prime }\right)}^2+xy{y}^{\prime \prime }-y{y}^{\prime }=0$
$\Rightarrow xy{y}^{\prime \prime }+x{\left(y^{\prime }\right)}^2-y{y}^{\prime }=0$
which is the required differential equation.