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(C) The equation of the curve is $y^n = a^{n-1}x$.Differentiating both sides with respect to $x$,we get:$n y^{n-1} \frac{dy}{dx} = a^{n-1}$Therefore,t

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$2$

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(C) The equation of the curve is $y^n = a^{n-1}x$.Differentiating both sides with respect to $x$,we get:$n y^{n-1} \frac{dy}{dx} = a^{n-1}$Therefore,the slope of the tangent at any point $(x_1, y_1)$ is:$\left( \frac{dy}{dx} \right)_{(x_1, y_1)} = \frac{a^{n-1}}{n y_1^{n-1}}$The length of the subnormal is given by the formula:$L = |y_1 \cdot \frac{dy}{dx}| = |y_1 \cdot \frac{a^{n-1}}{n y_1^{n-1}}| = |\frac{a^{n-1}}{n} y_1^{2-n}|$For the length of the subnormal to be constant,the expression must be independent of $y_1$. This happens when the exponent of $y_1$ is $0$.Thus,$2 - n = 0$,which implies $n = 2$.

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

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