Question

From the top of the tower h metre high , the angles of depression of two objects , which are in the line with the foot of the tower are $\alpha \mathrm{and}\beta (\beta >\alpha )$. Find the distance between the two objects .

Answer 1

Let the two objects be at points C and D.
The angle of depression for the point C is $\beta$ and for the point D is $\alpha$.
In ∆ABC,
$$\begin{gathered} \tan \beta =\frac{h}{BC} \\ \Rightarrow BC=\frac{h}{\tan \beta } \end{gathered}$$
In ∆ABD,
$$\begin{gathered} \tan \alpha =\frac{h}{BD} \\ \Rightarrow \tan \alpha =\frac{h}{BC+CD} \\ \Rightarrow \tan \alpha =\frac{h}{{\displaystyle \frac{h}{\tan \beta }}+CD} \\ \Rightarrow \frac{h}{\tan \beta }+CD=\frac{h}{\tan \alpha } \end{gathered}$$
$$\begin{gathered} \Rightarrow CD=\frac{h}{\tan \alpha }-\frac{h}{\tan \beta } \\ \Rightarrow CD=h\left[\frac{1}{\tan \alpha }-\frac{1}{\tan \beta }\right] \\ \Rightarrow CD=h\left[cot\alpha -cot\beta \right] \end{gathered}$$