Question : From two points on the ground and lying in a straight line through the foot of a pillar, the two angles of elevation of the top of the pillar are complementary. If the distances of the two points from the foot of the pillar are 12 metres and 27 metres and the two points lie on the same side of the pillar, then the height (in metres) of the pillar is:
Option 1: 12 metres
Option 2: 18 metres
Option 3: 15 metres
Option 4: 16 metres
Correct Answer: 18 metres
Solution :
Let the height of the pole AB be $h$ metre.
BC = 12 metres and BD = 27 metres
$\angle$ADB = $\theta$
Therefore, $\angle$ACB = $(90–\theta)$
From, $\Delta$ABC,
$\tan(90–\theta)=\frac{h}{12}$
⇒ $\cot\theta=\frac{h}{12}$ --------(1)
From, $\Delta$ABD,
$\tan\theta=\frac{h}{27}$ ------(2)
Multiplying equations (1) and (2), we have,
⇒ $h^{2}=324$
⇒ $h=18$ metres
Hence, the correct answer is 18 metres.
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