Normal PN bisects the angle S'PS.
Compare the triangles PNS and QPS'.
They are similar.
So
But
Hence
So the PSN is a right angled triangle.
The tangent at to the ellipse
cuts the y-axis at Q. The normal at P is parallel to the line joining Q to one focus S'. if S is the other focus, show that PS is parallel to the y-axis.
S'(-ae,0) and S(ae,0)
so gradient of QS'
this means
that would make ae=
but in order for PS to be vertical, ae needs to be
i.e. e=