Question

If $y^2=a{x}^2+bx+c$, where $a,b,c$ are constants, then $y^3\frac{d^{2}y}{d{x}^2}$ is equal to

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

Answer 1

Answer: (A)

a constant

Given, $y^2=a{x}^2+bx+c$
On differentiating w.r.t. $x$ we get
$2y\frac{dy}{dx}=2ax+b$
Again differentiating w.r.t. $x$ we get
$2{\left(\frac{dy}{dx}\right)}^2+2y\frac{d^{2}y}{d{x}^2}=2a$
$\Rightarrow y\frac{d^{2}y}{d{x}^2}=a-{\left(\frac{dy}{dx}\right)}^2$
$\Rightarrow y\frac{d^{2}y}{d{x}^2}=a-{\left(\frac{2ax+b}{2y}\right)}^2\Rightarrow y\frac{d^{2}y}{d{x}^2}=\frac{4a{y}^2-{(2ax+b)}^2}{4y^2}$
$\Rightarrow 4y^3\frac{d^{2}y}{d{x}^2}=4a(a{x}^2+bx+c)-(4a^2{x}^2+4abx+b^2)$
$\Rightarrow y^3\frac{d^{2}y}{d{x}^2}=4ac-b^2\Rightarrow y^3\frac{d^{2}y}{d{x}^2}=\frac{4ac-b^2}{4}=constant$