If the solution of the differential equation `(dy)/(dx ) =( ax+ 4) /( 2y + f)`represents a circle, then the value of a is:

Options

A. 2
B. `-2`
C. 3
D. `-4`

Answer

B

Explanation

We have,
`(dy)/(dx)=(ax+3)/(2y+f)rArr(ax+3)dx=(2y+f)dy`
On integrating, we obtain
`a(x^(2))/(2)+3x=y^(2)+fy+CrArr-(a)/(2)x^(2)+y^(2)-3x+fy+C=0`
This will represent a circle, if
`-(a)/(2)=1" "[because" Coeff. of "x^(2)="Coeff. of "y^(2)]`
`and,(9)/(4)+f^(2)-Cgt0" "["Using :"g^(2)+f^(2)-c gt0]`
`rArr" "a=-2 and 9+4f^(2)-4Cgt0`