From a height of h units, a man observes the angle of elevation as α and angle of depression as β of the top and bottom respectively of a tower of height H (> 4h). To what further height should be climb so that the values of angle of elevation and angle of depression get interchanged for the top and bottom of the tower?

From a height of h units, a man observes the angle of elevation as α and angle of depression as β of the top and bottom respectively of a tower of height H (> 4h). To what further height should be climb so that the values of angle of elevation and angle of depression get interchanged for the top and bottom of the tower? Correct Answer H - 2h units

h (tan α + tan β)/tan β = h’ (tan α + tan β)/tan α

∴ h’ = h/tan β × tan α

⇒ h cot β × tan α = H - h

∴ h’ = H - h

Increase in height = h’ - h = H - h - h = H - 2h units

Shortcut:

Since the distance between the building does not change

If the angle interchanges corresponding height portion of the tallest building should be interchanged

Length of AB in original case = Length of BD in new case

H - h = h’

∴ h’ - h = H - 2h units