The length of the subtangent at any point of | vedclass

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(B) Given the curve equation: $x^m y^n = a^{m+n}$.Taking the natural logarithm on both sides: $m \ln x + n \ln y = \ln(a^{m+n})$.Differentiating with

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Abscissa

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(B) Given the curve equation: $x^m y^n = a^{m+n}$.Taking the natural logarithm on both sides: $m \ln x + n \ln y = \ln(a^{m+n})$.Differentiating with respect to $x$: $\frac{m}{x} + \frac{n}{y} \frac{dy}{dx} = 0$.Thus,$\frac{dy}{dx} = -\frac{m}{n} \frac{y}{x}$.The length of the subtangent is given by the formula: $LST = \left| \frac{y}{dy/dx} \right|$.Substituting the value of $\frac{dy}{dx}$: $LST = \left| \frac{y}{(-\frac{m}{n} \frac{y}{x})} \right| = \left| -\frac{nx}{m} \right| = \frac{n}{m} |x|$.Since $\frac{n}{m}$ is a constant,the length of the subtangent is proportional to the abscissa $x$.

The length of the subtangent at any point of the curve $x^my^n = a^{m+n}$ is proportional to

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