Question

Let a, b, c be real numbers a $\ne$ 0. If $\alpha$ is a root of $a^2{x}^2$ + bx + c = 0, $\beta$ is a root of $a^2{x}^2$ - bx - c = 0 and 0 < $\alpha <\beta$, then the equation $a^2{x}^2$ + 2bx + 2c = 0 has a root $\gamma$ that always satisfies

• A.
• B.
• C.
• D.

Answer 1

The correct option is D

Since $\alpha and\beta$ are the roots of given equations. So we have $a^2{\alpha }^2+b\alpha +c=0and{a}^2{\beta }^2-b\beta -c=0$

Let f(x) = $a^2{x}^2+2bx+2c=0$

$Thenf\left(\alpha \right)=a^2{\alpha }^2+2b\alpha +2c=a^2{\alpha }^2+2(b\alpha +c)=a^2{\alpha }^2-2a^2{\alpha }^2=-a^2{\alpha }^2=-veandf\left(\beta \right)=a^2{\beta }^2+2(b\beta +c)=a^2{\beta }^2+2a^2{\beta }^2+2a^2{\beta }^2=3a^2{\beta }^2=+ve$

since $f\left(\alpha \right)andf\left(\beta \right)$ are of opposite signs, therefore by theory of equations there lies a root $\gamma$ of the equation f(x) = 0 between α and β i.e., $\alpha <\gamma <\beta$