First slide

Theory of equations

Question

If $a, b, c$ are real numbers in G.P. such that $a$ and $c$ are positive, then the roots of the equation $a x^{2} + b x + c = 0$

Moderate

A

are real and are in ratio $b : a c$
B

are real
C

are imaginary and are in ratio 1: $ω$ , where $ω$ is a complex cube root of unity
D

are imaginary and are in ratio - $1 : ω$

Solution

We have, $b^{2} = a c$
Let, $a, β$ be the roots of the equation $a x^{2} + b x + c = 0 .$ Then,
$\begin{array}{l} α, β = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a} \\ = α, β = \frac{- \sqrt{a c} \pm i \sqrt{3 a c}}{2 a} [∵ b^{2} = a c] \end{array}$
$= α = (- \frac{1}{2} + i \frac{\sqrt{3}}{2}) \sqrt{\frac{c}{a}}$ and, $β = (- \frac{1}{2} - i \frac{\sqrt{3}}{2}) \sqrt{\frac{c}{a}}$
$= α = ω \sqrt{\frac{c}{a}}$ and, $β = ω^{2} \sqrt{\frac{c}{a}} \Rightarrow α : β = 1 : ω$

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