Let $$x^k+y^k=a^k,(a,k > 0)$$ and
$$\dfrac{dy}{dx}+\left(\dfrac{y}{x}\right)^{1/3}=0$$, then $$k$$ is :
Correct option is B. $$\dfrac{2}{3}$$
$$x^k+y^k=a^k$$
differentiating w.r.t. x
$$kx^{k-1}+k.y^{k-1}\dfrac{dy}{dx}=0$$
$$\dfrac{dy}{dx}=-\dfrac{kx^{k-1}}{k.y^{k-1}}$$
$$\dfrac{dy}{dx}=-\left(\dfrac{x}{y}\right)^{k-1}$$
$$\dfrac{dy}{dx}+\left(\dfrac{x}{y}\right)^{k-1}=0$$
$$k-1=\dfrac{-1}{3}$$
$$\boxed{k=\dfrac{2}{3}}$$