Orthogonal trajectories of family of the curve x^2/3 + y^2/3 = a^2/3, where a is any arbitrary constant is

$$\begin{gathered} {\mathrm{x}}^{2/3}+{\mathrm{y}}^{2/3}={\mathrm{a}}^{2/3}; \\ \Rightarrow \frac{2}{3}{x}^{\raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}+\frac{2}{3}{y}^{\raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\frac{dy}{dx}=0 \\ replace \frac{dy}{dx} by -\frac{dx}{dy} \\ \Rightarrow x^{\raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}+y^{\raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$3 $}\right.}\left(-\frac{dx}{dy}\right)=0 \\ \Rightarrow {\mathrm{y}}^{1/3}\mathrm{dy}={\mathrm{x}}^{1/3}\mathrm{dx} \end{gathered}$$

integrating ${\mathrm{x}}^{4/3}-{\mathrm{y}}^{4/3}=\mathrm{c}$