You don't have to do all that.

Equation of the normal to the parabola

for the parameter is

Now the slope of the normal is =

So for any parametric point "t" on the given parabola,the slope of the normal to the parabola at that point is -1/t

Now we need the point Q having parameter t'.

Now as the normal at Q is perpendicular to that at P, (-1/t')(-1/t)=-1

So, t'=-1/t

so the coordinates of Q are merely

where