Solve this equation explicitly for $\beta$

Question

Solve this equation explicitly for $\beta$:
$$T_{12}=\frac{x_{11}+x_{22}}{1+|\beta|^2}\beta$$
$x_{11}$ and $x_{22}$ are positive real numbers where $x_{11}>x_{22}$ and $T_{12}$ and $\beta$ are complex numbers. and $|\beta|<1$

Up to know I've tried this:

sol1 = Solve[
 Subscript[T, 12] == (Subscript[x, 11] + Subscript[x, 22])/(1 + Abs[β]^2) β &&
  {Subscript[x, 11], Subscript[x, 22]} ∈ Reals &&
  {Subscript[T, 12], β} ∈ Complexes,
 β
]

But it gives a complicated solution whereas the answer should be
$$\beta=\frac{T_{12}}{|T_{12}|}\sqrt{\frac{x_{22}}{x_{11}}}$$ according to the scientific article that I'm studying.

Answer 1

Edited to provide greater detail

eq = Subscript[T, 12] == (Subscript[x, 11] + Subscript[x, 22])/(1 + Abs[β]^2) β

First determine Abs[β]

Simplify[ComplexExpand[Abs[#] & /@ eq, {Subscript[T, 12], β}, 
    TargetFunctions -> Abs], Subscript[x, 11] + Subscript[x, 22] > 0]
(* (1 + Abs[β]^2) Abs[Subscript[T, 12]] == Abs[β] (Subscript[x, 11] + Subscript[x, 22]) *)

Simplify[Solve[%, Abs[β]]] // Flatten
(* {Abs[β] -> (Subscript[x, 11] + Subscript[x, 22] - Sqrt[-4 Abs[Subscript[T, 12]]^2 + 
    (Subscript[x, 11] + Subscript[x, 22])^2])/(2 Abs[Subscript[T, 12]]), 
    Abs[β] -> (Subscript[x, 11] + Subscript[x, 22] + Sqrt[-4 Abs[Subscript[T, 12]]^2 + 
    (Subscript[x, 11] + Subscript[x, 22])^2])/(2 Abs[Subscript[T, 12]])}

and then β itself

Simplify[Solve[eq /. #, β] & /@ %] // Flatten
(* {β -> (Subscript[T, 12] (Subscript[x, 11] + Subscript[x, 22] - 
    Sqrt[-4 Abs[Subscript[T, 12]]^2 + (Subscript[x, 11] + Subscript[x, 22])^2]))
    /(2 Abs[Subscript[T, 12]]^2), 
    β -> (Subscript[T, 12] (Subscript[x, 11] + Subscript[x, 22] + 
    Sqrt[-4 Abs[Subscript[T, 12]]^2 + (Subscript[x, 11] + Subscript[x, 22])^2]))
    /(2 Abs[Subscript[T, 12]]^2)} *)