Question

If the length of the sub-tangent at any point to the curve $x{y}^n=a$ is proportional to the abscissa, then 'n' is ______.

• A. Any non-zero real number
• B. $2$
• C. $-2$
• D. $1$

Answer 1

The correct option is A Any non-zero real number

Length of the sub-tangent is given by $\frac{y}{\frac{dy}{dx}}$

For the curve $x{y}^n=a$, differentiating both sides w.r.t x, we have

$nx{y}^{n-1}\frac{dy}{dx}+y^n=0$

$\therefore \frac{dy}{dx}=\frac{-y}{nx}$

$\therefore$ sub-tangent $=\frac{y}{\frac{dy}{dx}}$

$=\frac{y}{\frac{-y}{nx}}$

$=-nx$

Thus, for any non-zero real number n, the length of sub-tangent will be proportional to the abscissa.