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

$\because y^2=a{x}^2+bx+c\therefore 2y\frac{dy}{dx}=2ax+b\Rightarrow 2y\frac{d^{2}y}{d{x}^2}+\frac{dy}{dx}.2\frac{dy}{dx}=2aor{y}^3\frac{d^{2}y}{d{x}^2}=a{y}^2-{\left(y\frac{dy}{dx}\right)}^2=a(a{x}^2+bx+c)-{(ax+\frac{b}{2})}^2=a^2{x}^2+abx+ac-a^2{x}^2-\frac{b^2}{4}-abx=ac-\frac{b^2}{4}=constant$