If a curve is represented parametrically by the equations $x=4 t^3+3$ and $y=4+3 t^4$ and $\frac{d^2 x}{d y^2} /\left(\frac{d x}{d y}\right)^n$ is constant then the value of $n$, is

5

We have, $x=4 t^3+3$ and $y=4+3 t^4$

$\frac{d x}{d t}=12 t^2$ and $\frac{d y}{d t}=12 t^3$

∴  $\frac{d y}{d x}=\frac{d y / d t}{d x / d t}=\frac{12 t^3}{12 t^2}=t$

$\Rightarrow \frac{d x}{d y}=\frac{1}{t}=t^{-1}$

$\Rightarrow \frac{d^2 x}{d y^2}=-t^{-2} \frac{d t}{d y}=-t^{-2} \times \frac{1}{12 t^3}=-\frac{1}{12} t^{-5}$

∴  $\frac{d^2 x}{d y^2} /\left(\frac{d x}{d y}\right)^n=-\frac{1}{12 t^5} \times t^n=-\frac{1}{12} t^{n-5}$

It is given that $\frac{d^2 x}{d y^2} /\left(\frac{d x}{d y}\right)^n$ is constant. Therefore, n = 5.