Question

If the quadratic equation $a{x}^2+bx+a^2+b^2+c^2-ab-bc-ca=0$, where $a,b,c$ are distinct real numbers, has imaginary roots, then

• A. $2(a-b)+{(a-b)}^2+{(b-c)}^2+{(c-a)}^2>0$
• B. $2(a-b)+{(a-b)}^2+{(b-c)}^2+{(c-a)}^2<0$
• C. $2(a-b)+{(a-b)}^2+{(b-c)}^2+{(c-a)}^2=0$
• D. none of these

Answer 1

The correct option is A $2(a-b)+{(a-b)}^2+{(b-c)}^2+{(c-a)}^2>0$

$a{x}^2+bx+a^2+b^2+c^2-ab-bc-ca=0$
$\Rightarrow 2a{x}^2+2bx+{(a-b)}^2+{(b-c)}^2+{(c-a)}^2=0$
Let $f\left(x\right)=2a{x}^2+2bx+{(a-b)}^2+{(b-c)}^2+{(c-a)}^2$
$f\left(0\right)={(a-b)}^2+{(b-c)}^2+{(c-a)}^2>0$
and $\Delta <0$ (Since, it is given that the roots are imaginary)
$\therefore f\left(x\right)>0$ ,$\forall x\in R$
$\therefore f(-1)>0$
$2a-2b+{(a-b)}^2+{(b-c)}^2+{(c-a)}^2>0$
$\Rightarrow 2(a-b)+{(a-b)}^2+{(b-c)}^2+{(c-a)}^2>0$