From  an  aero  plane  vertically  above  a  straight  horizontal  road,  the  angles  of depression of two consecutive milestones on opposite sides of the aero plane are observed to be α and β. Show that the height of the aero plane above the road is

Tan α tanβ/ tan α + tanβ

Ans: Let P Q be h

QB be x

Given : AB = 1 mile

QB = x

AQ = (1- x) mile in ?PAQ

Tan α PQ /AQ

Tan α = h/1 - x

1-x = h/Tan α                 .............1

In ΔPQB

Tan β = h/ x

x =  h/Tanβ

Substitute for x in equation (1)

1 = h/Tanβ  + h /Tanα

1/h = Tanβ + Tanα/TanβTanα

∴h = Tan α tanβ/ tan α + tanβ