Question

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 $\alpha and\beta$. Show that the height in miles of aeroplane above the road is given by $\frac{tan\alpha tan\beta }{tan\beta +tan\alpha }$

Answer 1

So from figure-

Let, $AB=h,BC=x$

So, in $△ABC$

We get ,

$\tan \alpha =\frac{h}{x}$...(i)

Similarly ,

In $△ABD$

$andtan\beta =\frac{h}{y}$...(ii)

We know that,

$BC+BD=CD=1\Rightarrow x+y=1$

using (1)& (2),

$x+y=\frac{h}{\tan \alpha }+\frac{h}{\tan \beta }=1$

$=>h=\frac{\tan \alpha \tan \beta }{\tan \alpha +\tan \beta }$

Hence, proved.