Let $ \alpha $ and $\beta $ be any two distinct complex numbers,then $|\alpha-\sqrt{\alpha^2-\beta^2}|+|\alpha+\sqrt{\alpha^2-\beta^2}|=$
My Attempt
Let $z_1=\alpha-\sqrt{\alpha^2-\beta^2}$,
$z_2=\alpha+\sqrt{\alpha^2-\beta^2}$
$z_1$ and $z_2$ are the roots of the complex valued equation
$z^2-2\alpha z+{\beta}^2=0$
So
$z_1+z_2=2\alpha$
and
$z_1z_2={\beta}^2$
now since alpha and beta are complex we cannot assume the roots to be conjugate and hence
$\overline{z_1}=z_2$ does not hold and consequently I am unable to find the necessary equations to solve for the value of
$|z_1|+|z_2|$
Please help me out!
Thanks in advance.