I am asked to find the minimum length confidence interval for a Beta distribution with parameters $a=\theta$ and $b=1$ and probability $1-\alpha$.
I have found that $-2\theta \sum_i \log(X_i)$ has distribution $\chi^2_{2n}$ and is therefore a pivotal quantity. However, I don't know how to find the minimum length confidence interval: I know I have to find two points $a, b$ such that $F(b) - F(a) = 1-\alpha$ and $f(b) = f(a)$ but I don't know how to tackle the problem numerically. Am I right on my approach until now? How should I approach the problem of solving for $b$ and $a$?
Thanks in advance.