If $α, β$ are the roots ${ax}^{2} + bx + c = 0$ and $k \in R$ . Then the condition so that $α < k < β$ is

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$a^{2} k^{2} + abk + ac < 0$

ac > 0

${ak}^{2} + bk + c > 0$

ac < 0

answer is D.

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Detailed Solution

$\begin{array}{l} α, β a r e r o o t s o f a x^{2} + b x + c = 0 a n d α < k < β \\ ∴ a f (k) < 0 \\ ∴ a (a k^{2} + b k + c) < 0 \\ \Rightarrow a^{2} k^{2} + a b k + a c < 0 \end{array}$

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