Question

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

$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1 ....$(i)
On differentiating Eq. (i) w.r.t. $x$, we get
$\frac{2 x}{a^2}+\frac{2 y}{b^2} \frac{d y}{d x} =0$
or $\frac{y}{x}\left(\frac{d y}{d x}\right) =\frac{-b^2}{a^2} .....$(ii)
On differentiating both sides Eq. (ii) w.r.t. x, we get
$\left(\frac{y}{x}\right)\left(\frac{d^2 y}{d x^2}\right)+\left(\frac{x \frac{d y}{d x}-y}{x^2}\right) \frac{d y}{d x}=0$
or $x y \frac{d^2 y}{d x^2}+x\left(\frac{d y}{d x}\right)^2-y \frac{d y}{d x}=0 .....$(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 $x^2+y^2 =2 a y ....$(i)
where, a is an arbitrary constant. Differentiating both sides of eq. (i) w.r.t. $x$, we get

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