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 ____.

Question

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 ____.

Accepted Answer

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

BM=x
   and

CM=y
  . Given that,

∠BAO=α
   and

∠CAN=β
  .From the figure, we have

∠ABM=α
   and

∠ACM=β
   since they are alternate angles of BAO and CAN respectively. In

ΔAMB
  , we have

tanθ=
      
       Perpendicular
      
       Base

tanα=
      
       AM
      
       BM

=
      h
      x

x=
    h
    
     tanα

… (1)In

ΔAMC
  , we have

tanθ=
      
       Perpendicular
      
       Base

tanβ=
      h
      y

y=
    h
    
     tanβ

… (2)Adding (1) and (2) to get:

x+y=
      h
      
       tanα
     
     +
      h
      
       tanβ

1=
      h
      
       tanα
     
     +
      h
      
       tanβ

h

1
        
         tanα
       
       +
        1
        
         tanβ

=1

h

tanα+tanβ
        
         tanαtanβ

=1

h=
      
       tanαtanβ
      
       tanα+tanβ

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

tanαtanβ
    
     tanα+tanβ

.

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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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 ____.