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Q.

A person observes the top P of a vertical tower OP of height h from a station S1 and finds that β1 is the angle of elevation. He moves in a horizontal plane to second station S2 and finds that PS2S1 is γ1 and the angle subtended by S2S1 at P is δ1 and the angle of elevation is β2. He moves again to a third station S3 such that S3S2=S2S1,  PS3S2=γ2 and the angle subtended by S3S2 at P is δ2. then 


sinγ1sinβ1sinδ1=sinγ2sinβ2sinδ2=hS1S2


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a

True

b

False 

answer is A.

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Detailed Solution

Sine rule in given as,
sinAa=sinBb=sinCc
Here,
a, b, c      sides of a triangle
A, B, C     Angles (lying opposite to sides a, b, c respectively.)
Use sine rule in the triangles PS1S2 and PS2S3 with the involvement of sides PS1S1 and S2S1 for triangle PS1S2 and sides PS2S2 and S3S2 for triangle PS2S3.
It is known that,
sinθ=PerpendicularHypotaneous
h=PS1sin β1​ 
PS1=cosec β1.
From ΔPS1S2  by sine formula,
PS1​​sinγ1​​​=S1S2sinδ1​​​
h​​sinβ1sinγ1​​​=S1S2sinδ1​​​ ……. (1)
h​​S1S2​​=sinβ1sinγ1sinδ1​​​
Question ImageIn the same way,
h​​S2S3​​=sinβ2sinγ2sinδ2​​​
But S2S3=S1S2   (given)
sinβ1sinγ1sinδ1​​​​​
=sinβ2sinγ2sinδ2​​​
=h​​S1S2​​
Option 1 is Correct.
 
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