If the length of the sub-tangent at any point | vedclass

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(D) The length of the sub-tangent at any point $(x, y)$ on a curve is given by the formula: $	ext{Length of sub-tangent} = \left| \frac{y}{dy/dx} i

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$y = x^{1/k} C$

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(D) The length of the sub-tangent at any point $(x, y)$ on a curve is given by the formula: $	ext{Length of sub-tangent} = \left| \frac{y}{dy/dx} ight|$.According to the problem,the length of the sub-tangent is proportional to the abscissa $x$. Let the constant of proportionality be $1/k$ (or simply $k$,but to match the options,we use $k$ as the proportionality constant such that $\frac{y}{dy/dx} = kx$).Thus,$\frac{y}{dy/dx} = kx$.Rearranging the terms to separate the variables,we get: $\frac{dy}{y} = \frac{1}{k} \frac{dx}{x}$.Integrating both sides: $\int \frac{dy}{y} = \frac{1}{k} \int \frac{dx}{x}$.This gives: $\ln|y| = \frac{1}{k} \ln|x| + \ln|C|$.Using logarithmic properties: $\ln|y| = \ln|x^{1/k}| + \ln|C| = \ln|C x^{1/k}|$.Therefore,the equation of the curve is $y = C x^{1/k}$.

If the length of the sub-tangent at any point $P$ on a curve is proportional to the abscissa of the point $P$,then the equation of that curve is ($C$ is an arbitrary constant).

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