Question

A flagstaff standing on the top of a tower subtends an angle $ta{n}^{-1}\left(1/9\right)$ at a point on the horizontal line through the foot of the tower where the tower subtends an angle $ta{n}^{-1}\left(4/5\right)$, then

Answer 1

Let $\angle PAQ=\alpha \Rightarrow \tan \alpha =\frac{1}{9}$
And $\angle PAO=\beta \Rightarrow \tan \beta =\frac{4}{5}$
Now in $△PAO$
$\frac{PO}{OA}=\tan \beta =\frac{4}{5}$ ...(1)
And $△QAO$
$\frac{QA}{OA}=\tan (\alpha +\beta )\Rightarrow \frac{h+PO}{OA}=\frac{\tan \alpha +\tan \beta }{1-\tan \alpha \tan \beta }\Rightarrow \frac{h+OQ}{OA}=1$
Substituting value from (1)
$\frac{OQ}{OA}=1-\frac{4}{5}=\frac{1}{5}$
Therefore if $OA=100\Rightarrow OQ=20\Rightarrow h=80AP=\sqrt{{OA}^2+{OQ}^2}=20\sqrt{41}$
Hence
Height of flagstaff $OQ=20m$
Hegiht of tower $h=80m$
Distance of point from foot of tower $OA=100m$
Distance of point from top of tower $AP=20\sqrt{41}m$