Two poles A B of length a metres and C D of length a + b ( b ≠ a ) metres are erected at the same horizontal level with bases at B and D . If B D = x and tan ∠ A C B = 1 2 , then:

Step-by-step solution

Let ∠ A B C = ∠ B C D = α ,   ∠ A C B = θ B D = A E = x A B = D E = a From Diagram tan ( θ + α ) = x b tan α = x a + b tan θ = 1 2 tan ( θ + α ) = tan θ + tan α 1 - tan θ tan α 1 2 + x a + b 1 − 1 2 ⋅ x a + b = x b a + b + 2 x 2 ( a + b ) − x = x b a b + b 2 + 2 b x = 2 a x + 2 b x − x 2 x 2 − 2 a x + a b + b 2 = 0 x 2 - 2 a x + b ( a + b ) = 0