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Height and Distance
Height and Distance
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Question

A tower T 1 of height 60 m is located exactly opposite to a tower T 2 of height 80 m on a straight road. Fromthe top of T 1 , if the angle of depression of the foot of T 2 is twice the angle of elevation of the top of T 2 , then the width (in m) of the road between the feetof the towers T 1 and T 2 is :

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Solution

Let the distance between T 1 and T 2 be x From the figure EA = 60 m (T 1 ) and DB = 80 m (T 2 ) DEC=θ\angle DEC = \theta and BEC=2θ\angle BEC = 2\theta Now in DEC\angle DEC, tanθ=DCAB=20x\tan \theta = {{DC} \over {AB}} = {{20} \over x} and in ΔBEC\Delta BEC, tan2θ=BCCE=60x\tan 2\theta = {{BC} \over {CE}} = {{60} \over x} We know that tan2θ=2tanθ1(tanθ)2\tan 2\theta = {{2\tan \theta } \over {1 - {{\left( {\tan \theta } \right)}^2}}} \Rightarrow 60x=2(20x)1(20x)2{{60} \over x} = {{2\left( {{{20} \over x}} \right)} \over {1 - {{\left( {{{20} \over x}} \right)}^2}}} \Rightarrow x2=1200{x^2} = 1200 \Rightarrow x=203x = 20\sqrt 3

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