The two curves $x=y^2$  and  $xy=a^3$  cut orthogonally at a point then a²

$$\begin{gathered} m_1{m}_2=-1 \\ \begin{array}{l}x=y^{2}xy=a^3\\ 2y{y}^1=1x{y}^1+y=0\\ m_1=\frac{1}{2y}{m}_2=\frac{-y}{x}\\ \begin{array}{l}{m}_1{m}_2=-1\\ \left(\frac{1}{2y}\right)\left(\frac{-y}{x}\right)=-1\Rightarrow x=\frac{1}{2},y=\frac{1}{\sqrt{2}}\Rightarrow xy=a^3\Rightarrow a^3=\frac{1}{2\sqrt{2}}\Rightarrow a^2=\frac{1}{2}\end{array}\end{array} \end{gathered}$$