Question

The set of the possible values of $a$ for which the expression $x^2-ax+1-2a^2$ is always positive is $(\alpha ,\beta )$, then

• A. $\alpha -2\beta =2$
• B. $\alpha +2\beta =2$
• C. $\beta +2\alpha =2$
• D. $\beta -2\alpha =2$

Answer 1

The correct option is A $\alpha -2\beta =2$

$(x^2-ax-2a^2+1)$ is always positive for $ax=(\alpha ,\beta )$

$\therefore D<O$

$\therefore a^2-4\left(1\right)(1-2a^2)<0$

$\therefore a^2-4+8a^2<0$

$\therefore 9a^2-4<0$

$\therefore a^2<\frac{4}{3}$

$\therefore a\in (\frac{-2}{3},\frac{2}{3})$

$\therefore \alpha =2/3$

$\beta =-2/3$

$\therefore \alpha /2\beta =\frac{2}{3}+\frac{4}{3}=2$