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Common Roots

If α, β, Y ar...

Question

If $α, β, γ$ are the roots of $x^{3} + 64 = 0$ , then the equation whose roots are ${(\frac{α}{β})}^{2} a n d {(\frac{α}{γ})}^{2}$ is

A

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B

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C

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D

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Solution

The correct option is B
Let $α, = 4, β = 4 ω, γ = 4 ω^{2}$
$∴ {(\frac{α}{β})}^{2} = \frac{1}{ω^{2}} = ω a n d {(\frac{α}{γ})}^{2} = \frac{1}{ω^{4}} = ω^{2}$
$∴$ Required equation is
$x^{2} - (ω - ω^{2}) x + ω^{3} = 0$
$\Rightarrow x^{2} + x + 1 = 0$

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