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 the coordinates of Q are merely