Question

Find the values of the parameter $a$ such that the roots $\alpha ,\beta$ of the equation $2x^2+6x+a=0$ satisfy the inequality $\frac{\alpha }{\beta }$ + $\frac{\beta }{\alpha }<2$.

• A. $a<0$
• B. $a<9/2$
• C. $a>9/2$
• D. $a>0$

Answer 1

Answer: (A), (C)

$a<0$
$a>9/2$

We have $\alpha +\beta =-3$ and $\alpha \beta =\frac{a}{2}$.
Now, $\frac{\alpha }{\beta }+\frac{\beta }{\alpha }<2$
or $\frac{{\alpha }^2+{\beta }^2}{\alpha \beta }<2$
or $\frac{(\alpha +\beta {)}^2-2\alpha \beta }{\alpha \beta }<2$
$\Rightarrow \frac{9-a}{\frac{a}{2}}<2$
or $\frac{9-a}{a}<1$
or $\frac{9-a}{a}-1<0$
or $\frac{2a-9}{a}>0$
$\Rightarrow a<0$ or $a>\frac{9}{2}$.