The abscissa of the point on the curve ${\mathrm{ay}}^2={\mathrm{x}}^3$, the normal at which makes equal intercepts on the axes is

normal makes equal intercepts on axes $\Rightarrow \frac{-1}{{\displaystyle \frac{\mathrm{dy}}{\mathrm{dx}}}}=\pm 1 \Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=\pm 1$

$\begin{array}{rcl}{\mathrm{ay}}^2& =& {\mathrm{x}}^3 \Rightarrow 2\mathrm{ay}\frac{\mathrm{dy}}{\mathrm{dx}}=3{\mathrm{x}}^2\end{array}$

$\begin{array}{rcl}\frac{\mathrm{dy}}{\mathrm{dx}}& =& 1 \Rightarrow \mathrm{y}=\frac{3{\mathrm{x}}^2}{2\mathrm{a}}\end{array}$  Substitute in given curve 

$\mathrm{a}\times \frac{9{\mathrm{x}}^4}{4{\mathrm{a}}^2}={\mathrm{x}}^3 \Rightarrow \mathrm{x}=\frac{4\mathrm{a}}{9}$