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Help finding x of hypergeometric 2F1[a,b,c,x] ?

Hello all,

The above function is given by Mathematica as
Hypergeometric2F1[a,b,c,d].  Define the ff,

h1 = 2F1[a,b,1,1-x]
h2 = 2F1[a,b,1,x]

Problem: Given a,b where 0<(a,b)<1, and a+b=1, find the unique real
number x with 0<x<1 such that,

     h1/h2 = sqrt[n]

for arbitrary rational n>0.

For certain (a,b) namely, (1/2, 1/2), (1/3, 2/3), (1/4, 3/4) and (1/6,
5/6), closed-form solutions are known for x. For example, the first
reduces to finding the "elliptic modulus k" and x can easily be given
as x = ModularLambda[Sqrt[-n]].  Surprisingly though, it seems little
is known for other (a,b).

Question: Is there Mathematica code to numerically evaluate x to
arbitrary precision for any (a,b) and n? The fact that x is 0<x<1
greatly helps. To illustrate, say for (1/5, 4/5) and given a certain n,
one can find x up to a few digits precision by plugging in values x =
0.1, 0.2,...0.9 and observing how close (h1/h2)^2 comes to n, and, say,
if it is between 0.4-0.5, then the values 0.41, 0.41,...0.50, and so
on, with x becoming increasingly precise as the "range" where x lies
becomes narrower. I did this manually (I know, crude) but there must be
code for this.

Any help will be appreciated.


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