# The angles of elevation of the top of from two points P and Q at distance $m^{2}$ and $n^{2}$ respectively, from the base and in the same straight line with it are complementary. The height of the tower is.

A. $(m n)^{\frac{1}{2}}$ B. $m n^{\frac{1}{2}}$ C. $m^{\frac{1}{2}} n$ D. $m n$ Answer: Option D
$\begin{array}{l} \tan \theta=\frac{H}{m^{2}} \\ \Rightarrow \tan \left(90^{\circ}-\theta\right)=\frac{H}{n^{2}} \\ \Rightarrow \cot \theta=\frac{H}{n^{2}} \\ \Rightarrow \tan \theta \cdot \cot \theta=\frac{H}{m^{2}} \times \frac{H}{n^{2}} \\ \Rightarrow \frac{H}{m^{2}} \times \frac{H}{n^{2}}=1 \\ \Rightarrow H^{2}=m^{2} n^{2} \\ \Rightarrow H=m n \end{array}$