Q)  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. Show that the height in miles of aeroplane above the road is given by tan α tan ẞ / (tan α + tan ẞ).

Ans: 

Step 1: Let’s draw a diagram for the given question:

Let AB or h be the height of the plane; , P and Q be the milestones and angles of depression be α and β respectively.

Also distance between PQ and is 1 mile (for being consecutive milestones). We need to find the height h.

Here, ∠ P will be equal to α and ∠ Q will be equal to β (for being, alternate interior angles)

Step 2: In Δ  ABP, tan P =

∴ tan α =

∴ PA = ………….. (i)

Step 3: In Δ  ABQ, tan Q =

∴ tan β =

∴ AQ = ………….. (ii)

Step 4: From the diagram, we can see that PQ = PA + AQ

By substituting values of PA & AQ from equations (i) and (ii), respectively, we get:

PQ = PA + AQ

∴ D =

∴ D =

∴ D =

Since it is given that D = 1

∴ 1 =

∴  h =

Hence Proved !

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