Question

From an aeroplane vertically above a straight horizontal road, the angles of depression of two consecutive milestones on opposite sides of the aeroplane are observed to be α and β. Show that the height in miles of the aeroplane above the road is given by `(tan alpha tan beta)/(tan alpha + tan beta)`

Answer 1

Let h be the height of aeroplane p above the road.

And A and B be the two consecutive milestones, then AB = 1  mile. we have ∠PAQ = α and ∠PBQ = β.

We have to prove

`h = (tan alpha tan beta)/(tan alpha  + tan beta)`

The corresponding figure is as follows

In ΔPAQ

`=> tan alpha = (PQ)/(AQ)`

`=> tan alpha = h/x`

`=> x = h/(tan alpha)`

`=> x = h cot alpha`

Again in ΔPBQ

`=> tan beta =(PQ)/(BQ)`

`=> tan beta = h/y`

`=> y = h/(tan beta)`

`=> y = h cot beta`

Now,

`=> AB  = x + y`

`=> AB = h(cot alpha  +  cot beta)`

`=> AB  = h(1/tan alpha + 1/tan beta)`

`=> AB = h((tan alpha + tan beta)/(tan alpha tan beta))`

Therefore `h = (tan alpha tan beta)/(tan alpha + tan beta)` (Since A B = 1)

Hence height of aero plane is  `(tan alpha tan beta)/(tan alpha + tan beta)`