Parabolic coordinates: express x,y and find kinetic energy

[metgt4]

Homework Statement

(ξ,η) in a plane are defined by η = (x² + y²)^1/2 - x and ξ = (x² + y²)^1/2 + x

Find x and y in terms of ξ and η. Show that the kinetic energy of a particle of mass m is:

T = (m/8)(ξ + η)(ξ'²/ξ + η'²/η)

The Attempt at a Solution

My attempt is scanned and appended in this post. I found x and y in terms of η and ξ, but am not sure where to go from there in order to find the kinetic energy in terms of the two. If somebody could point me in the right direction, I'd really appreciate it.

Thanks!
Andrew

 

[metgt4]

Nevermind, got it figured out. I went wrong in finding x and y as well.