First slide

Theory of equations

Question

If $0 < a < b < c$ , and the roots $α, β$ of the equation $a x^{2} + b x + c = 0$ are imaginary, then

Moderate

A

$| α | = | β |, | α | > 1$
B

$| α | \geq 1$
C

$| β | < 1$
D

none of these

Solution

It is given that the roots are imaginary. Therefore, $b^{2} - 4 a c < 0 .$ The roots $α$ and $β$ are given by
$α = \frac{- b + i \sqrt{4 a c - b^{2}}}{2 a}$ and $β = \frac{- b - i \sqrt{4 a c - b^{2}}}{2 a}$
Clearly, $α = \bar{β} .$ Therefore, $| α | = | β | .$
Further more,
$| α | = \sqrt{\frac{b^{2}}{4 a^{2}} + \frac{4 a c - b^{2}}{4 a^{2}}} = \sqrt{\frac{c}{a}} \Rightarrow | α | > 1 [∵ c > a]$

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