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Global Maxima

If α, β are...

Question

If $α, β$ are the non-zero roots of $a x^{2} + b x + c = 0$ and $α^{2}, β^{2}$ are the roots of $a^{2} x^{2} + b^{2} x + c^{2} = 0$ , then $a, b, c$ are in

A

G . P .

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B

H . P .

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C

A . P .

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D

none of these

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Solution

The correct option is B $G . P .$
From the sum and product property of quadratic equations, we have $α + β = (- b / a)$ ,
$α β = c / a$ from the first equation.
Now, from the second equation, we have $α^{2} + β^{2} = - {(b / a)}^{2}$ ie ${(α + β)}^{2} - 2 α β = - {(b / a)}^{2}$ .
Now, putting the respective values of $α + β$ and $α β$ , we get $b^{2} = a c$ which is the required condition for $G . P$ .

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