Question

If $x^m+y^m=1$ such that $\frac{dy}{dx}=-\frac{x}{y}$, then what should be the value of $m$?

• A. $0$
• B. $1$
• C. $2$
• D. None of the above

Answer 1

The correct option is C $2$

Given : $\Rightarrow \frac{dy}{dx}=-\frac{x}{y}$ ...... $\left(i\right)$

$x^m+y^m=1$

Differentiating with respect to $x$ we get:

$m{x}^{m-1}+m{y}^{m-1}\frac{dy}{dx}=0$

$\Rightarrow \frac{dy}{dx}=-{\left(\frac{x}{y}\right)}^{m-1}$

$\Rightarrow -{\left(\frac{x}{y}\right)}^{m-1}=-\frac{x}{y}$ ... [From $\left(i\right)$]

$\Rightarrow m-1=1\therefore m=2$

Hence, option C is correct.