Two poles AB of length a metres and CD of length a + b ( b ≠ a )...

Let $∠ ABC = ∠ BCD = \alpha, ∠ ACB = \theta$

$BD = AE = x$

$AB = DE = a$

From Diagram

$\tan ( \theta + \alpha ) = \dfrac{x}{b}$

$tan\alpha = \dfrac{x}{a + b}$

$tan\theta = \dfrac{1}{2}$

$\tan ( \theta + \alpha ) = \dfrac{tan\theta + tan\alpha}{1 - tan\theta tan\alpha}$
$\dfrac{\dfrac{1}{2} + \dfrac{x}{a + b}}{1 - \dfrac{1}{2} ⋅ \dfrac{x}{a + b}} = \dfrac{x}{b}$

$\dfrac{a + b + 2x}{2 ( a + b ) - x} = \dfrac{x}{b}$

$ab + {b}^{2} + 2bx = 2ax + 2bx - {x}^{2}$

${x}^{2} - 2ax + ab + {b}^{2} = 0$

${x}^{2} - 2ax + b ( a + b ) = 0$