Length of the subtangent at (x_{1}, y_{1}) on | vedclass

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(B) Given the curve $x^{n} y^{m}=a^{m+n}$,where $m, n > 0$. Taking the natural logarithm on both sides: $n \ln x + m \ln y = (m+n) \ln a$. Differentia

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$\frac{m}{n}|x_{1}|$

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(B) Given the curve $x^{n} y^{m}=a^{m+n}$,where $m, n > 0$. Taking the natural logarithm on both sides: $n \ln x + m \ln y = (m+n) \ln a$. Differentiating with respect to $x$: $\frac{n}{x} + \frac{m}{y} \frac{dy}{dx} = 0$. Solving for the derivative: $\frac{dy}{dx} = -\frac{n}{m} \cdot \frac{y}{x}$. The length of the subtangent is defined as $|\frac{y}{dy/dx}|$. Substituting the derivative: $|\frac{y}{-(n/m)(y/x)}| = |-\frac{m}{n} x| = \frac{m}{n} |x|$. Thus,at the point $(x_{1}, y_{1})$,the length of the subtangent is $\frac{m}{n} |x_{1}|$.

Length of the subtangent at $(x_{1}, y_{1})$ on $x^{n} y^{m}=a^{m+n}, m, n>0$,is

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