Let $λ \neq 0$ be a real number. Let $α, β$ be the roots of the equation $14 x^{2} - 31 x + 3 λ = 0$ and $α, γ$ be the roots of the equation $35 x^{2} - 53 x + 4 λ = 0$ . Then $\frac{3 α}{β} and \frac{4 α}{γ}$ are the roots of the equation

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$7 x^{2} - 245 x + 250 = 0$

$49 x^{2} - 245 x + 250 = 0$

$7 x^{2} + 245 x - 250 = 0$

$49 x^{2} + 245 x + 250 = 0$

answer is B.

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Detailed Solution

$\begin{array}{l} 14 α^{2} - 31 α + 3 λ = 0 \\ 35 α^{2} - 53 α + 4 λ = 0 by eliminating λ \Rightarrow α = \frac{5}{7} \\ α + β = \frac{31}{14} \\ β = \frac{31}{14} - \frac{5}{7} = \frac{21}{14} = \frac{3}{2} \\ γ = \frac{53}{35} - \frac{5}{7} = \frac{53 - 25}{35} = \frac{28}{35} = \frac{4}{5} \\ x^{2} - (\frac{3 \cdot \frac{5}{7}}{\frac{3}{2}} + \frac{4 . \frac{5}{7}}{\frac{4}{5}}) ϕ + \frac{12 \cdot \frac{25}{49}}{\frac{3}{2} \cdot \frac{4}{5}} = 0 \\ x^{2} - (\frac{10}{7} + \frac{25}{7}) x + \frac{250}{49} = 0 \\ 49 x^{2} - 245 x + 250 = 0 \end{array}$