If  $y^2=a{x}^2+bx+c$, then  $y^3.\frac{d^{2}y}{d{x}^2}$ is 

$y^2=a{x}^2+bx+c$

$2y.\frac{dy}{dx}=2ax+b$

$2y.\frac{d^{2}y}{d{x}^2}+\frac{dy}{dx}.2\frac{dy}{dx}=2a$

$\Rightarrow y\frac{d^{2}y}{d{x}^2}+2{\left(\frac{dy}{dx}\right)}^2=a$

$y^3\frac{d^{2}y}{d{x}^2}+y^2{\left(\frac{dy}{dx}\right)}^2=a{y}^2$

$y^3.\frac{d^{2}y}{d{x}^2}=a{y}^2-{\left(y\frac{dy}{dx}\right)}^2$

$=a(a{x}^2+bx+c)-{(ax+\frac{b}{2})}^2$

$=ac-\frac{b^2}{4}=$constant