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Geometric Progression

Let α, β be...

Question

Let $α, β$ be root of the quadratic equation $x^{2} k x + (k^{2} + 2 k - 4) = 0$ , then the maximum value of $α^{2} + β^{2}$ is equal to :

8

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\frac{4}{9}

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\frac{16}{9}

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\frac{8}{9}

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Solution

The correct option is A $8$
$\begin{matrix} α + β = k α β = k^{2} + 2 k - 4 α^{2} + β^{2} = {(α + β)}^{2} - 2 α β = k^{2} - 2 (k^{2} + 2 k - 4) = - k^{2} - 4 k + 8 = f (k) f^{'} (k) = - 2 k - 4 = 0 \to k = - 2 f (- 2) = - {(- 2)}^{2} - 4 x - 2 + 8 = 12 max i m u m v a l u e o f α^{2} + β^{2} = 8. \end{matrix}$

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