Question

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

• A. a constant
• B. a function of x only
• C. a function of y only
• D. a function of x and y

Answer 1

The correct option is A

a constant

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

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

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

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

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

= $a^2{x}^2+abx+ac-a^2{x}^2-\frac{b^2}{4}-abx$

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