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 alpha and beta. Show that the height of the aeroplane above the road is tan alpha*tan beta/tan alpha +tan beta

Answer 1

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Answer 2

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$\huge\fcolorbox{black}{aqua}{Solution:-}$

✏️ Given:-

★ 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 alpha and beta.

✏️ To show:-

★ h = tan α.tan β /tan α + tan β

✏️ Answer:-

★ Let h be the height of aeroplane 'A' above the road and B & C be two consecutive milestones. Then,

BC = 1 mile

Now,

In ∆ADB, we have

AD/BD = tan α

⇒h/BD = tan α

⇒BD = h/tan α

In ∆ADC, we have

AD/DC = tan β

⇒h/DC = tan β

⇒h/DC = tan β

⇒DC = h/tan β

⇒BC = BD + DC = h/tan α + h/tan β = h [1/tan α + 1/tan β]

⇒1 = h [tan β + tan α/tan α.tan β]

[ ∵ BC = 1 mile ]

∴ h = (tan α.tan β)/tan α + tan β

Hence Proved! ^^

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