Orthogonal trajectory of family of curve $x^{\frac{2}{3}}+y^{\frac{2}{3}}=a$ where $a$ is arbitrary constant is

$\frac{2}{3}{x}^{\frac{2}{3}-1}-\frac{2}{3}{y}^{\frac{2}{3}-1}\frac{dy}{dx}=0$

Now replace $\frac{dy}{dx}$ by $\frac{dx}{dy}$ and then solve the differential equation to get orthogonal  curve