A particle moves on the parabola $y^2 = 4ax$ in such a way that its projection on the y-axis has a constant velocity. Then its projection on x-axis moves with

we have, 

$\begin{array}{l}{y}^2=4ax\\ \Rightarrow 2y\frac{dy}{dt}=4a\frac{dx}{dt}\\ \Rightarrow y\frac{dy}{dt}=2a\frac{dx}{dt}\\ \Rightarrow y\frac{d^{2}y}{d{t}^2}+{\left(\frac{dy}{dt}\right)}^2=2a\frac{d^{2}x}{d{t}^2}\end{array}$

$\Rightarrow y\times 0+(\text{ Constant }{)}^2=2a\frac{d^{2}x}{d{t}^2}$                 $\left[\because \frac{dy}{dt}=\text{ Constant }\right]$

$\Rightarrow \frac{d^{2}x}{d{t}^2}=$ Constant  

$\Rightarrow$Projection $(x, 0)$ of any point $(x, y)$ on X-axis moves with constant acceleration.