Let α and β are the roots of equation a x 2 + b x + c = 0 a ≠ 0 . If 1 , α + β , α β are in arithmetic progression and α , 2 , β are in harmonic progression, then the value of α 2 + β 2 - 2 α 2 β 2 2 α 2 + β 2 is equal to

Question

Let α and β are the roots of equation a x 2 + b x + c = 0 a ≠ 0 . If 1 , α + β , α β are in arithmetic progression and α , 2 , β are in harmonic progression, then the value of α 2 + β 2 - 2 α 2 β 2 2 α 2 + β 2 is equal to, 0, 0.5, 1, 1.5

MARKS App · Accepted Answer

1 , α + β , α β are in A.P. ⇒ 1 , - b a , c a are in A.P. ⇒ 1 + c a = - 2 b a ⇒ a + c + 2 b = 0 … 1 1 α , 1 2 , 1 β are in A.P. ⇒ 1 α + 1 β = 1 ⇒ α + β = α β ⇒ - b a = c a ⇒ b + c = 0 … 2 From 1 & 2 we get, a = - b = c ⇒ α , β are roots of equation x 2 - x + 1 = 0 Now, α 2 + β 2 - 2 α 2 β 2 2 α 2 + β 2 = 1 2 - ( α β ) 2 ( α + β ) 2 - 2 α β = 1 2 - 1 2 1 2 - 2 1 = 1 2 + 1 = 1.5

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