Question

If the solution of the differential equation $\frac{dy}{dx}=\frac{ax+3}{2y+f}$ represents a circle, then the value of ${}^{\prime }{a}^{\prime }$ is

• A. 2
• B. -2
• C. 3
• D. -4

Answer 1

Answer: (B)

We have,
$\frac{dy}{dx}=\frac{ax+3}{2y+f}$
$\Rightarrow (ax+3)dx=(2y+f)dy$
$\Rightarrow a\frac{x^2}{2}+3x=y^2+fy+C$ (Integrating)
$\Rightarrow -\frac{a}{2}{x}^2+y^2-3x+fy+C=0$
This will represent a circle, if
$-\frac{a}{2}=1$ $[\because$ Coeff. of $x^2=$ Coeff. of $y^2]$
and,$\frac{9}{4}+\frac{f^2}{4}-C>0$
$[$ Using $:g^2+f^2-c>0$ ]
$\Rightarrow a=-2$ and $9+f^2-4C>0$