Question : From two points, lying on the same horizontal line, the angles of elevation of the top of the pillar are $\theta$ and $\phi$ ($\theta<\phi$). If the height of the pillar is $h$ m and the two points lie on the same sides of the pillar, then the distances between the two points are:
Option 1: $h(\tan\theta-\tan\phi)$ metre
Option 2: $h(\cot\phi-\cot\theta)$ metre
Option 3: $h(\cot\theta-\cot\phi)$ metre
Option 4: $h\frac{(\tan\theta \tan\phi)}{(\tan\phi-\tan\theta)}$ metre
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Correct Answer: $h(\cot\theta-\cot\phi)$ metre
Solution :
Let AB = height of pole = $h$ metre
$\angle$ACB = $\theta$, $\angle$ADB = $\phi$
In ∆ABD,
$\tan\phi=\frac{AB}{BD}$
⇒ $BD=h\cot\phi$
In ∆ABC,
$\tan\theta=\frac{AB}{BC}$
⇒ $BC=h\cot\theta$
∴ Required distance, CD
$=h\cot\theta-h\cot\phi$
$= h(\cot\theta-\cot\phi)$ metre
Hence, the correct answer is $h(\cot\theta-\cot\phi)$ metre.
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