it's late and i cant get these 2 question to fall out! help please

I have point on ellipse

I have the tangent and normal to this ellipse meet Y-axis at T and N. Then i found the centre of the circle through T,N,P to be

The last part asks for the equation of the locus of Q on PX produced such that X is the midpoint of PQ

so ive let Q=(x,y) then

so

but i cant find the locus. The book says

i cant get this with my Q. Maybe im making a mistake or my Q is wrong or the book is wrong......

2)

Also having trouble with this:

the tangent at to

ellipse

cuts the Y-axis at Q. The normal at P is parallell to the line joining Q to one focus S'. If S is the other focus, show that PS is parallel to the y-axis.

found Q to be

focus with

grad

but QS parallel to normal so grad normal= grad QS

so

and grad

now im assuming i use the grad QS=grad normal to show the denominator of grad PS=0 but i cant seem to get it....