A flagstaff stands vertically on a pillar, the height of the...

Let,& $AB$ is a pillar and $BC$ is the flagstaff on it and man is standing at $D$
We have, $\dfrac{h}{x} = tan \alpha = a$ (say) (in $∆ ABD$ )
$\tan 2\alpha = \dfrac{4h}{x} = 4a$ (in $∆ ACD$ )
$⇒ \dfrac{2tan \alpha}{1 - \tan^{2} \alpha} = 4a$
$⇒ \dfrac{2a}{1 - a^{2}} = 4a ⇒ 2 = 4 - 4a^{2} ⇒ 4a^{2} = 2 ⇒ a = \dfrac{1}{2}$
$∴ h: x = 1: \sqrt{2}$