All the points on the curve $y^{2}=4 a|x+a \sin (x / a)|$, where the tangent is parallel to the axis of $x$ are lies on

$y^{2}=4 a\left[x+a \sin \left(\frac{x}{a}\right)\right]$...(i) $\therefore \quad 2 y \frac{d y}{d x}=4 a\left[1+\cos \left(\frac{x}{a}\right)\right]$...(ii) If tangent is parallel to $x$ -axis, then $\frac{d y}{d x}=0$ So, from Eq. (i), we get $\cos \left(\frac{x}{a}\right)=-1$ $\therefore$ $$ \sin \left(\frac{x}{a}\right)=0 $$ On putting this value in Eq. (i), we get $y^{2}=4 a(x+0) \Rightarrow y^{2}=4 a x$ So, all the points on the curve $$ y^{2}=4 a\left(x+a \sin \frac{x}{a}\right) $$ where the tangent is parallel to the $x$ -axis are lies on parabola.