Parametric coordinates of parabola? Can $(a(\sin(t))^2 , 2a \sin(t))$ be the parametric coordinates of the parabola $y^2 = 4ax$ ?
I found that these coordinates satisfy the equation of the parabola but my friend says that although these satisfy the equation , these can not be the coordinates. Can any body give me an explanation for this?

The coordinates of the point are periodic, and share a period. They satisfy the equation of the parabola, and they are continuous. You can therefore expect that as $t$ increases, the point will trace out some portion of the parabola over and over.
Specifically (assuming $a>0$), if $t$ increases from $0$, the point will move from $(0,0)$ up and to the right until it reaches $(a,2a)$ when $t=\frac\pi2$. As $t$ continues to increase, the motion will reverse and the point will arrive back at the origin when $t=\pi$. It will continue through the origin and move down and to the right until it reaches $(a,-2a)$ when $t=\frac{3\pi}2$. Then it will reverse and return to the origin, reaching $(0,0)$ when $t=2\pi$. After that, everything repeats. The point continues to swing up and down along the same part of the parabola as $t$ increases without bound.
