Question

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

• A. ${(abscissa)}^2$
• B. ${(abscissa)}^3$
• C. abscissa
• D. ordinate

Answer 1

Answer: (C)

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 $=\left|y\frac{dx}{dy}\right|=\frac{nx}{m}\propto x$