Question

Length of the subtangent at $ (x_1, y_1)$ on $x^n y^m = a^{m+n}, m, n > 0,$ is

• A. $\frac {n}{m}|x_1|$
• B. $\frac {m}{n}|x_1|$
• C. $\frac {n}{m}|y_1|$
• D. $\frac {m}{n}|y_1|$

Answer 1

Answer: (B)

Given, $x^{n}y^{m} = a^{m + n}, m, n >\, 0$
Taking logarithm on both sides, we get
log $\left(x^{n}y^{m}\right) = log \,a^{m + n}$
$\Rightarrow log x^{n} + log\, y^{m} = \left(m + n\right)$ log a
$\Rightarrow n log x + m log y = \left(m + n\right) log a\quad ... \left(i\right)$
On differentiating Eq. (i) w.r.t. 'x', we get
$\frac{n}{x}+\frac{m}{y} = 0$
$\Rightarrow \frac{m}{y} \frac{dy}{dx} = -\frac{n}{x}$
$\Rightarrow \frac{dy}{dx} = -\left(\frac{n}{m}\right)\left(\frac{y}{x}\right)$
$\therefore $ Length of subtangent
$= \frac{y}{dy / dx}$
$= \frac{y}{-\left(\frac{n}{m}\right)\left(\frac{y}{x}\right)}$
$= \frac{-mx}{n}$
$\therefore $ Length of sub tangent at $\left(x_{1}, y_{1}\right) = \frac{m}{n}\left|x_{1}\right|$