એક સ્તંભએ ત્રિકોણીય બાગ \(ABC\) ની અંદર શિરોલંબ દિશામાં રાખેલ છે. જો સ્તંભની ટોચનો દરેક ખૂણાથી ઉસ્ત્ધેધકોણ \(\frac{\pi}{3}\) અને \(\Delta ABC\) ની પરિત્રિજ્યા \(2\) હોયતો સ્તંભની ઊંચાઈ મેળવો.

A \(\frac{2 \sqrt{3}}{3}\)
B \(2 \sqrt{3}\)
C \(\sqrt{3}\)
D \(\frac{1}{\sqrt{3}}\)

Verified Solution

Answer & Solution

Correct Answer

(B) \(2 \sqrt{3}\)

Step-by-step Solution

Detailed explanation

Let \(P D=h, R=2\) As angle of elevation of top of pole from \(A , B , C\) are equal \(So\) \(D\) must be circumcentre of \(\triangle ABC\) \(\tan \left(\frac{\pi}{3}\right)=\frac{ PD }{ R }=\frac{ h }{ R }\) \(h=R \tan \left(\frac{\pi}{3}\right)=2 \sqrt{3}\)

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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