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From an aeroplane vertically above a straight horizontal road, the angles of depression of two consecutive mile stones on opposite sides of the aeroplane are observed to be $α$ and $β$ . The height in miles of aeroplane above the road is given by ____.

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Detailed Solution

Suppose AM is the aeroplane's height above the ground, stone 1 is at B, and stone 2 is at C. Take

B M = x

and

C M = y

. Given that,

∠ B A O = α

and

∠ C A N = β

From the figure, we have

∠ A B M = α

and

∠ A C M = β

since they are alternate angles of BAO and CAN respectively.
In

Δ A M B

, we have

tan θ = P e r p e n d i c u l a r B a s e tan α = A M B M = h x

x = h tan α

… (1)
In

Δ A M C

, we have

tan θ = P e r p e n d i c u l a r B a s e tan β = h y

y = h tan β

… (2)
Adding (1) and (2) to get:

x + y = h tan α + h tan β 1 = h tan α + h tan β h 1 tan α + 1 tan β = 1 h tan α + tan β tan α tan β = 1 h = tan α tan β tan α + tan β

Hence, the height in miles of aeroplane above the road is given by

tan α tan β tan α + tan β

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