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Given $\alpha_j$, $\beta_j$ real (positive or negative) with $j=1,2$ and $M>1$ real, the following

$\sqrt{(M^2-1)\alpha_j^2-\beta_j^2}$ have to be real, so, $\left |\beta_j\right |<\left | \alpha_j\right |\sqrt{M^2-1}$

What can be said about $\beta_1-\beta_2$? I know the answer but can't get it on my own.

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    You can use this : $|\beta_1-\beta_2| \le |\beta_1| + |\beta_2|.$ – 2017-02-17
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    Could you elaborate on that? Thanks – 2017-02-17
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    It's triangle inequality. Because $|x-y|=|x+(-y)| \le |x|+|-y| = |x|+|y|.$ Are you clear now? – 2017-02-17

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