For the curve \(4 x^{5}=5 y^{4}\), the ratio of the cube of the subtangent at a point on the curve the square of the subnormal at the same point is

The given curve is \(4 x^{5}=5 y^{4}\)
\[
\begin{gathered}
\Rightarrow \quad 20 x^{4}=20 y^{3} \cdot \frac{d y}{d x} \\
\Rightarrow \quad \frac{d y}{d x}=\frac{x^{4}}{y^{3}}
\end{gathered}
\]
We know that
Length of subnormal \((S N)=\left(y \cdot \frac{d x}{d y}\right)=\left(\frac{y^{4}}{x^{4}}\right)\)
Length of subtangent \((\mathrm{ST})=\left(\mathrm{y} \cdot \frac{\mathrm{dy}}{\mathrm{dx}}\right)=\left(\frac{\mathrm{x}^{4}}{\mathrm{y}^{2}}\right)\)
But given condition is
\[
\begin{aligned}
\frac{(\mathrm{SN})^{3}}{(\mathrm{ST})^{2}}=\frac{\left(\mathrm{y}^{4} / \mathrm{x}^{4}\right)^{3}}{\left(\mathrm{x}^{4} / \mathrm{y}^{2}\right)^{2}}=\left(\frac{\mathrm{y}^{4}}{\mathrm{x}^{4}}\right)^{3} \times\left(\frac{\mathrm{y}^{2}}{\mathrm{x}^{4}}\right)^{2} \\
=\frac{\mathrm{y}^{12}}{\mathrm{x}^{12}} \times \frac{\mathrm{y}^{4}}{\mathrm{x}^{8}}=\left(\frac{\mathrm{y}^{16}}{\mathrm{x}^{20}}\right) \\
=\left(\frac{\mathrm{y}^{4}}{\mathrm{x}^{5}}\right)^{4} \\
=\left(\frac{4}{5}\right)^{4}=\frac{4^{4}}{5^{4}} \quad\left(\because 4 \mathrm{x}^{5}=5 \mathrm{y}^{4}\right)
\end{aligned}
\]