Q:

$-ia(1\pm \sqrt{1-1/a^2})$, $a>0$ inside unit circle?

Given $a>0$ I would like to know whether:
$\alpha=-ia(1+ \sqrt{1-1/a^2})$ and
$\beta =-ia(1- \sqrt{1-1/a^2})$ are inside the unit circle.
How can I check that?

A:

You have
$$\alpha\cdot\beta = (-i)^2a^2\left(1 - \sqrt{1-1/a^2}^2\right) = -a^2(1/a^2) = -1,$$
so either both lie on the unit circle, or one lies inside the unit disk and the other outside.
For $0 < a \leqslant 1$, the radicand $1-1/a^2$ is a non-positive real number, hence the square root purely imaginary, and then $\lvert\alpha\rvert = \lvert\beta\rvert = 1$.
For $a > 1$, the radicand $1-1/a^2$ is positive, and if you take the branch of the square root that is positive on the positive real axis, you have $\lvert\beta\rvert < \lvert\alpha\rvert$, and hence then $\beta$ lies inside the unit disk, $\alpha$ outside.

