Let α and β be the roots of x 2 - 3 x + p = 0 and γ and δ be the roots of x 2 - 6 x + q = 0 . If α , β , γ , δ from a geometric progression. Then ratio ( 2 q + p ) : ( 2 q - p ) is

Step-by-step solution

Let α = a ,   β = a r ,   γ = a r 2 ,   δ = a r 3 α + β = 3    ⇒ a + a r = 3 ........... 1 γ + δ = 6   ⇒ a r 2 + a r 3 = 6 ......... 2 By 1 and 2 ⇒   a r 2 1 + r a 1 + r = 6 3 ⇒   r 2 = 2 ∴ 2 q + p 2 q − p = 2 γ δ + α β 2 γ δ − α β = 2 a 2 r 5 + a 2 r 2 a 2 r 5 − a 2 r = 2 r 4 + 1 2 r 4 − 1 = 8 + 1 8 − 1 = 9 7