The sub tangent at any point on the curve xm yn=am+n varies as

The sub tangent at any point on the curve $x^{m} y^{n} = a^{m + n}$ varies as

A
(abscissa)2
B
(ordinate)2
C
abscissa
D
ordinate

Solution:

We have,

$x^{m} y^{n} = a^{m + n}$

$\Rightarrow m \log_{e} x + n \log_{e} y = (m + n) \log_{e} a$

$\Rightarrow \frac{m}{x} + \frac{n d y}{y} \frac{d y}{d x} = 0$ [Differentiating w.r.t. x]

$\Rightarrow \frac{d y}{d x} = - \frac{m y}{n x}$

$∴$ Length of the subtangent $= |\frac{y}{d y}| = |y \times \frac{n x}{∣ d x}| = \frac{n}{m} | x | \propto x$

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