The minimum distance from the origin to a point on the curve ${\mathrm{x}}^{2/3}+{\mathrm{y}}^{2/3}={\mathrm{a}}^{2/3}(\mathrm{a}>0)$ is

$$\begin{gathered} {\mathrm{x}}^{2/3}+{\mathrm{y}}^{2/3}={\mathrm{a}}^{2/3} \\ \mathrm{point} \mathrm{on} \mathrm{the} \mathrm{carve} \mathrm{is} \\ \mathrm{P}\left(\mathrm{a} {\cos }^3\mathrm{\theta }, \mathrm{a} {\sin }^3\mathrm{\theta }\right) \\ 0\left(0,0\right) \mathrm{in} \mathrm{origin} \end{gathered}$$

$$\begin{gathered} {\mathrm{OP}}^2={\mathrm{a}}^2{\cos }^6\mathrm{\theta }+{\mathrm{a}}^2{\sin }^6\mathrm{\theta } \\ ={\mathrm{a}}^2\left(1-3 {\sin }^2\mathrm{\theta } {\cos }^2\mathrm{\theta }\right) \\ ={\mathrm{a}}^2\left(1-\frac{3}{4}{\sin }^{2}2\mathrm{\theta }\right) \\ {\mathrm{OP}}^2 \min \Rightarrow {\sin }^{2}2\mathrm{\theta } \max . \\ \Rightarrow {\sin }^{2}2\mathrm{\theta }=1 \\ {\mathrm{OP}}^2={\mathrm{a}}^2\left(1-\frac{3}{4}\right)=\frac{{\mathrm{a}}^2}{4} \\ \mathrm{OP}=\frac{\mathrm{a}}{2} \end{gathered}$$