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Question

The angles A, B, C of a ∆ABC are in AP and the sides a, b, c are in G.P. If a2 + c2 = λb2, then λ = ____________.

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Solution

Since A, B, C are A.P
⇒ 2B = A + C
Since A + B + C = π (By angle sum property)
⇒ 3B = π
i.e B=π3 ...1
also,
Since a, b, c are in g.p
⇒ b2 = ac ....(2)
Using cosB = a2+c2-b22ac

i.e cosπ3 = a2 + c2 - ac2ac from (1) and (2).

12=a2 + c2 -ac2ac
⇒ ac = a2 + c2 - ac
⇒ a2 + c2 = 2ac
⇒ a2 + c2 = 2b2 from (2)
Hence λ = 2

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