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If the subnormal at any point on the curve \(y^n=a x\) is constant then the value of \(n\) is
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Equation of given curve \(y^n=a x\), so
\(n y^{n-1} \frac{d y}{d x}=a \Rightarrow \frac{d y}{d x}=\frac{a}{n} y^{1-n}\)
\(\therefore\) The length of subnormal \(=\left|y \frac{d y}{d x}\right|=\left|y \frac{a}{n} y^{1-n}\right|\)
\(=\left|\frac{a}{n} y^{2-n}\right|\)
It is given that length of subnormal is constant, means it must be independent of \(x\) and \(y\) terms, so \(n=2\).
Hence, option (b) is correct.
\(n y^{n-1} \frac{d y}{d x}=a \Rightarrow \frac{d y}{d x}=\frac{a}{n} y^{1-n}\)
\(\therefore\) The length of subnormal \(=\left|y \frac{d y}{d x}\right|=\left|y \frac{a}{n} y^{1-n}\right|\)
\(=\left|\frac{a}{n} y^{2-n}\right|\)
It is given that length of subnormal is constant, means it must be independent of \(x\) and \(y\) terms, so \(n=2\).
Hence, option (b) is correct.
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