Question

If $0<a<b<c$ and the roots $\alpha ,\beta$ of the equation $a{x}^2+bx+c=0$ are imaginary, then

• A. $\mid \alpha \mid =\mid \beta \mid$
• B. $\mid \alpha \mid >1$
• C. $\mid \alpha \mid <1$
• D. $\alpha =\mid \beta \mid$

Answer 1

Answer: (A), (B)

The correct options are
A $\mid \alpha \mid =\mid \beta \mid$
B $\mid \alpha \mid >1$

Given, $\alpha ,\beta$ are the roots of the equation $a{x}^2+bx+c=0$
Since, the roots are imaginary, therefore $b^2-4ac<0$.
The roots $\alpha$ and $\beta$ are given by
$\alpha =\frac{-b+i\sqrt{4ac-b^2}}{2a}$ and $\beta =\frac{-b-i\sqrt{4ac-b^2}}{2a}$

Clearly, $\alpha =\overline{\beta }$.
Therefore, $\mid \alpha \mid =\mid \beta \mid$.
and $\mid \alpha \mid =\sqrt{\frac{b^2}{4a^2}+\frac{4ac-b^2}{4a^2}}=\sqrt{\frac{c}{a}}$
$\Rightarrow \mid \alpha \mid >1[\because c>a]$
Hence, options 'A' and 'B' are correct.