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
$\frac{tan α tan β}{tan α + tan β}$

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Solution

Let $B$ and $C$ be the two consecutive mile stones.
$∴$ $B C = B D + C D = 1 m i l e s$

Let the height of the aeroplane $A D = H m i l e s$
$\Rightarrow$ In $△ A B D, tan α = \frac{A D}{B D}$ ........ $t a n (Θ) = \frac{O p p o s i t e}{A d j a c e n t}$

$\Rightarrow$ $tan α = \frac{h}{B D}$

$\Rightarrow$ $B D = \frac{h}{tan α}$ -------- ( 1 )

$\Rightarrow$ In $△ A C D, tan β = \frac{A D}{C D}$

$\Rightarrow$ $tan β = \frac{h}{C D}$ .......... $t a n (Θ) = \frac{O p p o s i t e}{A d j a c e n t}$

$\Rightarrow$ $C D = \frac{h}{tan β}$ -------- ( 2 )

$\Rightarrow$ $B C = B D + C D = \frac{h}{tan α} + \frac{h}{tan β}$

$\Rightarrow$ $B C = h (\frac{1}{tan α} + \frac{1}{tan β})$

$\Rightarrow$ $1 = h (\frac{tan α + tan β}{tan α tan β})$

$∴$ $h = \frac{tan α tan β}{tan α + tan β}$
$\Rightarrow$ Hence, height of the aeroplane is $\frac{tan α tan β}{tan α + tan β}$

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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 βtan α + tan β

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
$\frac{tan α tan β}{tan α + tan β}$