Which of the following differential equation represents the family of circles touching the x-axis at the origin?

$\left(x^2-y^2\right) d y-2 x y d x=0$

faculty of circles touching x-axis at origin

equation of this circle will be

$-x^2+(y-a)^2=a^2$

$x^2+y^2+a^2-2ay=a^2$

$\Rightarrow x^2+y^2-2 a y=0$        .......(1)

differentiating w.r.t x

$\frac{d}{d x}\left(x^2+y^2-2 a y\right)=\frac{d}{d x}(0)$

$2 x+2 y \frac{d y}{d x}-2 a \frac{d y}{d x}=0$        ....(2)

from (1)

$x^2+y^2=2 a y$

$\Rightarrow \frac{x^2}{y}+\frac{y^2}{y}=2 a$

$\Rightarrow \frac{x^2}{y}+y=2a$

substituting value of 2a in eq (2)

⇒  $2 x+2 y \frac{d y}{d x}-\left(\frac{x^2}{y}+y\right) \frac{d y}{d x}=0$

$2 x+\frac{d y}{d x}\left(2 y-y-\frac{x^2}{y}\right)=0$

$2 x+\frac{d y}{d x}\left[y-\frac{x^2}{y}\right]=0$

$2 x y+\frac{d y}{d x}\left(y^2-x^2\right)=0$     (Multiplying both sides with y)

$\left(x^2-y^2\right) d y-2 x y d x=0$     (Multiplying both sides with (-dx))