Sorry for the multiple posts. I had to post them separately because it exceeded the maximum size.
First, let's find the gradient of PQ. That's going to be
By difference of two squares and cancelling common terms. What does this tell us? p+q is a constant value. Not sure how this helps us yet; let's find out.
We want the locus of N, which has co-ordinates
Hmm, actually, looks like you basically solved it! We want to show that it is a normal to the parabola. So, let's find the derivative.
by rearranging the original equation. At the point x=2km
Therefore, the negative reciprocal (normal) will be -a/km, which is the gradient you found in your working! Does that make sense? A bit of a weird, all over the place question. I'm in a rush so I hope this answer makes sense; if someone else wants to elaborate, feel free to!