$\begin{array}{l} T h e q u a d r a t i c e q u a t i o s x^{2} - 6 x + a = 0 a n d x^{2} - c x + 6 = 0 h a v e \\ o n e r o o t i n c o m m o n . T h e o t h e r r o o t s o f t h e f i r s t a n d \sec o n d e q u a t i o n s a r e int e g e r s i n t h e r a t i o 4 : 3. T h e n t h e c o m m o n r o o t i s \end{array}$

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answer is D.

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Detailed Solution

$\begin{array}{l} L e t α, 4 β a r e r o o t s o f x^{2} - 6 x + a = 0 \\ α + 4 β = - 6, 4 α β = a \\ a n d α, 3 β a r e r o o t s o f x^{2} - c x + 6 = 0 \\ α + 3 β = c, 3 α β = 6 \Rightarrow α β = 2 \\ ∴ a = 4 (2) \Rightarrow 8 \\ ∴ T h e E q u a t i o n x^{2} - 6 x + 8 = 0 \\ \Rightarrow (x - 2) (x - 4) = 0 \\ \Rightarrow x = 2, 4 \\ I f α = 4, t h e n β = \frac{1}{2} \\ i n t h i s c a s e 3 β i s n o t int e g e r \\ ∴ T h e c o m m o r o o t w i l l b e α = 2 \end{array}$