If a curve ` y=f(x) ` passes through the point `(1,-1)` and satisfies the differential equaiton .`y (1 + xy ) dx = x dy ` , then `f (-1/2)` is equal to :

Options

A. `-(2)/(5)`
B. `-(4)/(5)`
C. `(2)/(5)`
D. `(4)/(5)`

Answer

D

Explanation

The differential equation is
`y(1+xy)dx=xdy`
`rArr" "ydx-xdy=-xy^(2)dx`
`rArr" "(ydx-xdy)/(y^(2))=-xdx`
`rArr" "d((x)/(y))=-xdx`
On integrating, we obtain
`(x)/(y)=-(x^(2))/(2)+C`
It is given that the curve given by (i) passes though the point `(1,-1)`
`therefore" "-1=-(1)/(2)+CrArr C=-(1)/(2)`
Putting `C=-(1)/(2)` in (i), we obtain
`y(x^(2)+1)+2x=0`
Putting `x=-(1)/(2)` in (ii), we obtain `y=(4)/(5)`. Hence, `f(-(1)/(2))=(4)/(5)`