Inverting parabolic and stereographic coordinates

[Epimetheus]

Homework Statement

Given the parabolic co-ordinate system defined, given Cartesian coordinates x and y, as
$\mu=2xy$
$\lambda = x^2-y^2,$
find the inverse transformation x(\mu, \lambda) and y(\mu,\lambda).

Homework Equations

None

The Attempt at a Solution

We compute \lambda/\mu^2 in order to obtain
x=\pm \sqrt{\frac{\lambda\pm\sqrt{\lambda^2+1}}2}
and
y=\frac{2\mu}{\pm \sqrt{\frac{\lambda\pm\sqrt{\lambda^2+1}}2}}

However, I don't think this is right ... I'm following a set of CM lecture notes, which immediately go on to claim x^2+y^2=\lambda^2+\mu^2 and
\dot x^2 + \dot y^2 = \frac14 \frac{\dot \lambda^2 + \dot \mu^2}{\sqrt{\lambda^2 + \mu^2}}
which looks tantalizingly similar to spherical polars, but doesn't seem to follow from what I have ...

Any thoughts on inverting general "nonlinear" co-ordinate systems (other than cylindrical and spherical)? What about finding the inverse transformation for the stereographic projection defined by
\frac{x}{\xi} = \frac{y}{\eta} = \frac1{1-\zeta}

so that x=\frac{\xi}{1-\zeta}; y=\frac{\eta}{1-\zeta}?

It's hard to get rid of the "old" set of variables in this case.

 

[chaoseverlasting]

If you want to find x and y in terms of \lambda and \mu, then youre given \mu=2xy

y=\frac{\mu}{2x}.

Substitute this in the other equation and solve for x. Similarly for y.