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
$\Rightarrow nlogx+mlogy=\left(m+n\right)loga...\left(i\right)$
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 $\left(x_1,y_1\right)=\frac{m}{n}\left|x_1\right|$