Question

Let $a, b, c \in R$ and $a \neq 0$. If $\alpha$ is a root of $a^2 x^2+b x+c=0, \beta$ is a root $a^2 x^2-b x-c=0$ and $0<\alpha<\beta$, then the equation $a^2 x^2+2 b x+2 c=0$ has a root $\gamma$ that always satisfies

• A. $\alpha<\gamma<\beta$
• B. $\gamma=\alpha+\frac{\beta}{2}$
• C. $\gamma=\alpha$
• D. $\gamma=\frac{\alpha+\beta}{2}$

Answer 1

Answer: (A)

$\text { Let } f(x)=a^2 x^2+2 b x+2 c $
$f(\alpha)=a^2 \alpha^2+2 b \alpha+2 c=-a^2 \alpha^2$
$f (\beta)= a ^2 \beta^2+2 \beta b+2 c =3 a ^2 \beta^2 $
$\therefore f (\alpha) \cdot f (\beta)<0$
Hence, $\gamma$ lies between $\alpha$ and $\beta$.