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(C) Given the curve equation: $y^n = a^{n-1}x$. Differentiating both sides with respect to $x$: $n y^{n-1} \frac{dy}{dx} = a^{n-1}$. Thus,$\frac{dy}{d

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$2$

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(C) Given the curve equation: $y^n = a^{n-1}x$. Differentiating both sides with respect to $x$: $n y^{n-1} \frac{dy}{dx} = a^{n-1}$. Thus,$\frac{dy}{dx} = \frac{a^{n-1}}{n y^{n-1}}$. The length of the subnormal is given by the formula: $L = |y \frac{dy}{dx}|$. Substituting the value of $\frac{dy}{dx}$: $L = |y \cdot \frac{a^{n-1}}{n y^{n-1}}| = |\frac{a^{n-1}}{n} \cdot y^{1-(n-1)}| = |\frac{a^{n-1}}{n} \cdot y^{2-n}|$. For the length of the subnormal to be constant,the expression must be independent of $y$. This occurs when the exponent of $y$ is zero: $2 - n = 0$,which implies $n = 2$.

If the length of the subnormal at any point of the curve $y^n = a^{n-1}x$ is constant,then $n$ equals-

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