If the tangent at ‘p’ on the curve $x^{m} y^{n} = a^{m + n}$ meets the axes at A and B then $A P : P B =$

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$n : m$

$1 : 2$

$m : n$

$1 : 1$

answer is B.

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Detailed Solution

$\begin{aligned} x^{m} y^{n} = a^{m + n} \Rightarrow \log (x^{m} y^{n}) = \log a^{m + n} \\ \Rightarrow m \log x + n \log y = \log a^{m + n} \\ \frac{d}{d x} {m \log x + n \log y} = \frac{d}{d x} \{\log a^{m + n}\} \\ \Rightarrow \frac{m}{x} + \frac{n d y}{y} \frac{d y}{d x} = 0 \Rightarrow \frac{d y}{d x} = \frac{- m y}{n x} \end{aligned}$
$∴ Slope of the tangent of P (x_{1}, y_{1}) is$
$= {[\frac{dy}{dx}]}_{p} = \frac{- {my}_{1}}{{nxx}_{1}}$
$Equation of the tangent at P (x_{1}, y_{1}) is y - y_{1} = m (x - x_{1})$
$y - y_{1} = \frac{- m y_{1}}{n x_{1}} (x - x_{1})$
$\begin{aligned} \Rightarrow n x_{1} y - n x_{1} y_{1} = - m x_{1} + m x_{1} y_{1} \\ \Rightarrow m x_{1} + n x_{1} y = x_{1} y_{1} (m + n) \\ \Rightarrow \frac{m x y_{1}}{x_{1} y_{1} (m + n)} + \frac{n x_{1} y}{x_{1} y_{1} (m + n)} = 1 \\ \Rightarrow \frac{x}{\frac{x_{1} (m + n)}{m}} + \frac{y}{\frac{y_{1} (m + n)}{n}} = 1 \end{aligned}$
$Let the tangent cuts x -axis at A and y -axis at B .$
$\begin{aligned} ∴ O A = \frac{x_{1} (m + n)}{m}, O B = \frac{y_{1} (m + n)}{n} \end{aligned} A (\frac{x_{1} (m + n)}{m}, 0) B (0, \frac{y_{1} (m + n)}{n}) P (x_{1}, y_{1}) d i v i d e s A B i n t h e r a t i o A P : P B = x - x_{1} : x_{2} - x = n : m$