First slide

Theory of equations

Question

If the ratio of the roots of the equation $a x^{2} + b x + c = 0$ is equal to the ratio of the roots of the equation $x^{2} + x + 1 = 0,$ then $a, b, c$ are in

Moderate

A

AP.
B

G.P.
C

H.P.
D

none of these

Solution

Let $α, β$ be the roots of $a x^{2} + b x + c = 0 .$
Clearly, $ω$ and $ω^{2}$ are the roots of $x^{2} + x + 1 = 0 .$
$∴$ $\frac{α}{β} = \frac{ω}{ω^{2}} \Rightarrow β = α ω$
Now,
$α + β = - b / a$ and $α β = c / a$
$\Rightarrow α + α ω = - \frac{b}{a}$ and $α^{2} ω = \frac{c}{a}$
$\Rightarrow α ω^{2} = \frac{b}{a}$ and $α^{2} ω = \frac{c}{a}$
$\Rightarrow α = \frac{b}{a}$ and $α^{2} ω = \frac{c}{a}$
$\Rightarrow {(\frac{b}{a} ω)}^{2} ω = \frac{c}{a} \Rightarrow b^{2} = a c \Rightarrow a, b, c$ are in G.P.

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