Question

Consider the following statements
I. The differential equation representing the family of ellipses having foci on $X$-axis and centre at origin is
$xy\frac{d^{2}y}{d{x}^2}+x{\left(\frac{dy}{dx}\right)}^2-y\frac{dy}{dx}=0$.
II. The differential equation of the family of circles touching the $X$-axis at origin is $\frac{dy}{dx}=-\frac{2xy}{x^2-y^2}$.
Choose the correct option.

• A. Only I is true
• B. Only II is true
• C. Both I and II are true
• D. Neither I nor II is true

Answer 1

Answer: (A)

I. We know that the equation of said family of ellipse is

$\frac{x^2}{a^2}+\frac{y^2}{b^2}=\mathrm{1....}$(i)
On differentiating Eq. (i) w.r.t. $x$, we get
$\frac{2x}{a^2}+\frac{2y}{b^2}\frac{dy}{dx}=0$
or $\frac{y}{x}\left(\frac{dy}{dx}\right)=\frac{-b^2}{a^2}.....$(ii)
On differentiating both sides Eq. (ii) w.r.t. x, we get
$\left(\frac{y}{x}\right)\left(\frac{d^{2}y}{d{x}^2}\right)+\left(\frac{x\frac{dy}{dx}-y}{x^2}\right)\frac{dy}{dx}=0$
or $xy\frac{d^{2}y}{d{x}^2}+x{\left(\frac{dy}{dx}\right)}^2-y\frac{dy}{dx}=\mathrm{0.....}$(iii)
which is the required differential equation.
II. Let $C$ denote the family of circles touching $X$-axis at origin. Let $(0,a)$ be the coordinates of the centre of any member of the family. Therefore, equation of family $C$ is
$x^2+(y-a{)}^2=a^2$
or $x^2+y^2=2ay....$(i)
where, a is an arbitrary constant. Differentiating both sides of eq. (i) w.r.t. $x$, we get

$2x+2y\frac{dy}{dx}=2a\frac{dy}{dx}$
$\Rightarrow x+y\frac{dy}{dx}=a\frac{dy}{dx}$
$\Rightarrow a=\frac{x+y\frac{dy}{dx}}{\frac{dy}{dx}}.....$(ii)
On substituting the value of a from Eq. (ii) in Eq. (i), we get
$x^2+y^2=2y\frac{\left[x+y\frac{dy}{dx}\right]}{\frac{dy}{dx}}$
$\Rightarrow \frac{dy}{dx}\left(x^2+y^2\right)=2xy+2y^2\frac{dy}{dx}$
$\Rightarrow \frac{dy}{dx}=\frac{2xy}{x^2-y^2}$
This is the required differential equation of the given family of circles.