If the solution of differential equation $\frac{dy}{dx}=\frac{ax +3}{2y+5}$ represents a circle, then $a$ is equal to:

-2

The correct answer is Option (2) → -2

$\frac{dy}{dx}=\frac{ax +3}{2y+5}$

$⇒\int (2y+5)dy=\int (ax +3)dx$

$⇒y^2+5y=\frac{a}{2}x^2+3x+C$

for the equation to represent a circle,

$x^2+y^2+bx+Ey+F=0$

$≡y^2+5y-\frac{a}{2}x^2-3x-C=0$

$⇒-\frac{a}{2}=1$

$⇒a=-2$