Question

The solution of $\frac{dy}{dx}=\left(\frac{ax+b}{cy+d}\right)$ represents a parabola if :-

• A. $a=0,c=0$
• B. $a=1,c=2$
• C. $a=0,c\ne 0$
• D. $a=1,c=1$

Answer 1

The correct option is C $a=0,c\ne 0$

$\frac{dy}{dx}=\left(\frac{ax+b}{cy+d}\right)$
$\Rightarrow (cy+d)dy=(ax+b)dx$
$\frac{{cy}^2}{2}+dy=\frac{{ax}^2}{2}+bx+K$,

Intergrating both the sides

($K$ being the constant of integration)
The equation represents a parabola
If $c=0,a\ne 0ora=0,c\ne 0$

As we know that we can have two types of forms for the parabola equation

$(x-h{)}^2=4a(y-k\left)or\right(y-k{)}^2=4a(x-h)$ which can be achieved by taking the above conditions.