*pokes title*

The question is:

P is any point on the hyperbola with the equation

. S is the focus

and S' is the focus

, where e is the eccentricity. Show that

Mr F says: This is wrong. It should be 2ae. You can see this for yourself by taking P(a, 0) and P'(-a, 0) ...... It's ....
My working:

I've found the length ofSP and S'P using good old Pythagoras

, so

and similarly

, which gives me

If one of the signs changes in SP or S'P then I'd get the right answer. Can someone please check if I made a mistake with the signs? Thanks.