I solved Question 5.
First, simplify the question.
wlog let S=(0,a), L=(2a,a), A=(0,0), X=(0,-a) then it has equation x^2=4ay
rtp AN=NX i.e., N=(0,-a/2)
For the new parabola, vertex=(0,-a), focus=(0,a), focal length=2a
Then the equation is y=x^2/(4(2a))-a, so x^2=8a(y+a), directrix y=-3a...