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Q.
The sub tangent at any point of the curve $ {{x}^{m}}{{y}^{n}}={{a}^{m+n}} $ varies as
Jharkhand CECEJharkhand CECE 2012
Solution:
We have, $ {{x}^{m}}{{y}^{n}}={{a}^{m+n}} $
Taking logarithm on both sides, we get
$ \Rightarrow m\log x+n\log y=(m+n)\log a $
Differentiating both sides w.r.t. $ x, $ we get
$ \therefore $ $ \frac{m}{x}+\frac{n}{y}\frac{dy}{dx}=0 $
$ \Rightarrow $ $ \frac{dx}{dy}=-\frac{nx}{my} $
$ \therefore $ Subtangent $ =\left| y\frac{dx}{dy} \right|=\frac{nx}{m}\propto x $