If three distinct numbers a, b, c are in G.P. and the equations ax^2 + 2bx + c = 0 and dx^2 + 2ex + f = 0 have a common root, then which one of the following statements is correct?

Correct option is C. Let the common root be α.
Since a, b, c are in G.P., b² = ac.
Since α is a root of ax² + 2bx + c = 0, we have aα² + 2bα + c = 0.
Since α is a root of dx² + 2ex + f = 0, we have dα² + 2eα + f = 0.
From aα² + 2bα + c = 0, we have aα² + 2bα + c = 0. Since b² = ac, this can be written as (√a α + √c)² = 0, implying α = -√c/√a = -√c/√a.
From dα² + 2eα + f = 0, we can write this as d(-√c/√a)² + 2e(-√c/√a) + f = 0, which simplifies to dc/a - 2e√c/√a + f = 0.
Multiplying by a, we get dc - 2e√ac + fa = 0.
Since b² = ac, we have dc - 2eb + fa = 0.
This can be rewritten as da + fa = 2eb, which implies that da, eb, fc are in A.P.
Therefore, the correct statement is that da, eb, fc are in A.P.