A horizontal park is in the shape of a triangle with . A vertical lamp post is erected at the point such that and , where is the midpoint of . Then is equal to :

...... (i) ...... (ii) ....... (iii)

A horizontal park is in the shape of a triangle with . A ver... | JEE Main 2024 Height and Distance

Q: JEE Main 2024 Height and Distance: A horizontal park is in the shape of a triangle with . A vertical lamp post is erected at the point ...

The correct answer is A. ...... (i) ...... (ii) ....... (iii)

Question

A horizontal park is in the shape of a triangle $\mathrm{OAB}$ with $\mathrm{AB}=16$ . A vertical lamp post $\mathrm{OP}$ is erected at the point $\mathrm{O}$ such that $\angle \mathrm{PAO}=\angle \mathrm{PBO}=15^{\circ}$ and $\angle \mathrm{PCO}=45^{\circ}$ , where $\mathrm{C}$ is the midpoint of $\mathrm{AB}$ . Then $(\mathrm{OP})^{2}$ is equal to :

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$OP = OA\tan 15 = OB\tan 15$ ...... (i) $OP = OC\tan 45 \Rightarrow OP = OC$ ...... (ii) $OA = OB$ ....... (iii) $O{C^2} + {8^2} = O{A^2}$ $O{P^2} + 64 = O{P^2}{\left( {{{\sqrt 3 + 1} \over {\sqrt 3 - 1}}} \right)^2}$ $64 = O{P^2}\left[ {{{{{\left( {\sqrt 3 + 1} \right)}^2} - {{\left( {\sqrt {3 - 1} } \right)}^2}} \over {{{\left( {\sqrt 3 - 1} \right)}^2}}}} \right]$ $= O{P^2}\left( {{{4\sqrt 3 } \over {{{\left( {\sqrt 3 - 1} \right)}^2}}}} \right)$ $O{P^2} = {{64{{\left( {\sqrt 3 - 1} \right)}^2}} \over {4\sqrt 3 }} = {{32} \over {\sqrt 3 }}\left( {2 - \sqrt 3 } \right)$

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