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Question

# The angle of elevation of the top of a TV tower from three points A,B and C in a straight line through the foot of the tower are α,2α and 2α respectively. If AB=a, the height of the tower is

A
atanα
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B
asinα
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C
asin2α
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D
asin3α
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Solution

## The correct option is C asin2αBased on the given information, we can draw the figure shown above.Here, AB=aE is the top of the TV tower Let BD=x and ED=h be the height of the tower.We know that, tanθ=Opposite SideAdjacent Side∴ In ΔBDE, tan2α=hx⇒x=htan2αAlso, in ΔADE, tanα=ha+x⇒tanα=ha+htan2α We know that, tan2θ=2tanθ1−tan2θ∴ tanα=ha+h(1−tan2α)2tanα⇒tanα=2htanα2atanα+h−htan2α ⇒h(1+tan2α)=2atanα ⇒h=2asinαcosα ⇒h=asin2α ..... [∵ sin2θ=2sinθcosθ]Hence, the height of the TV tower is asin2α.

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