Let y(x) be a solution of `xdy+ydx+y^(2)(xdy-ydx)=0` satisfying y(1)=1.
Statement -1 : The range of y(x) has exactly two points.
Statement-2 : The constant of integration is zero.

Options

A. Statement-1 is True, Statement-2 is True, Statement-2 is a correct explanation for Statement-1.
B. Statement-1 is True, Statement-2 is True, Statement-2 is not a correct explanation for Statement-1.
C. Statement-1 is True, Statement-2 is False.
D. Statement-1 is False, Statement-2 is True.

Answer

C

Explanation

We have,
`xdy+ydx+y^(2)(xdy-ydx)=0`
`rArr" "d(xy)+x^(2)y^(2){(xdy-ydx)/(x^(2))}=0`
`rArr" "(1)/((xy)^(2))d(xy)+d((y)/(x))=0`
On integrating, we get
`-(1)/(xy)+(y)/(x)=C" …(i)"`
It is given that y = 1 when x = 1
`therefore" "-1+1=C rArr C=0`
Putting C = 0 in (i), we get
`-(1)/(xy)+(y)/(x)=0 rArr y^(2)-1=0 rArr y = pm1`
Clearly, statement-1 and statement-2 are true.
Also, statement-2 is not a correct explanation for statement-1.