I have a complex variable X of the form X=A*cos(theta)+i*A*sin(theta)
where A and theta are the amplitude and phase of X respectively.

I have an equation relating X to a real variable Y:

X=(a-b+i*a*b*Y)/(a+b+i*a*b*Y) (equation 1)

where a and b are real constants.

If I plot out the locus of X as Y varies, I can use geometry to prove the following relationship between the amplitude (A) and phase (theta) of X:

A=(1/(1+a/b))*cos(theta) +or- ((1/(1+a/b))*(cos(theta))^2-((1-a/b)/(1+a/b)))^0.5

I desperately need to prove this algebraically from equation 1, but am a lowly engineer and so am unable to. Any help achieving this would be greatly apreciated.