Let $y(x)$ be a solution of $x d y+y d x+y^2(x d y-y d x)=0$ satisfying $y(1)=1$.

Statement-1: The range of $y(x)$ has exactly two points.

Statement-2: The constant of integration is zero.

Statement-1 is True, Statement-2 is True; Statement-2 is not a correct explanation for Statement-1.

We have,

$x d y+y d x+y^2(x d y-y d x)=0$

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

$\Rightarrow \frac{1}{(x y)^2} d(x y)+d\left(\frac{y}{x}\right)=0$

On integrating, we get

$-\frac{1}{x y}+\frac{y}{x}=C$           ....(i)

It is given that $y=1$ when $x=1$

∴  $-1+1=C \Rightarrow C=0$

Putting $C=0$ in (i), we get

$-\frac{1}{x y}+\frac{y}{x}=0 \Rightarrow y^2-1=0 \Rightarrow y= \pm 1$

Clearly, statement-1 and statement-2 are true.

Also, statement- 2 is not a correct explanation for statement- 1 .