Question

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

• A. $(\text{abscissa}{)}^2$
• B. $(\text{ordinate}{)}^2$
• C. Abscissa
• D. Ordinate

Answer 1

Answer: (C)

Abscissa

We have,
$x^m{y}^n=a^{m+n}$
$\Rightarrow m{\log }_{e}X+n{\log }_{e}y=(m+n){\log }_{e}a$
On differentiating both sides w.r.t. $x$, we get
$\Rightarrow \frac{m}{x}+\frac{n}{y}\cdot \frac{dy}{dx}=0$
$\Rightarrow \frac{dy}{dx}=-\frac{my}{nx}$
Therefore, length of the subtangent $=\left|\frac{y}{dy/dx}\right|$
$=|-y\times \frac{nx}{my}|$
$=\frac{n}{m}|x|\propto x$