Question

The angles of elevation of the top of a tower at two points, which are at distances $a$ and $b$ from the foot in the same horizontal line and on the same side of the tower, are complementary. The height of the tower is

• A. $ab$
• B. $\sqrt{ab}$
• C. $\sqrt{\frac{a}{b}}$
• D. $\sqrt{\frac{b}{a}}$

Answer 1

The correct option is B

$\sqrt{ab}$

Step 1: Consider the following figure that represents the problem

Suppose $AB$ is the tower and $C$ and $D$ are the points of observations on $AC$.

Step 2: Determine the height from point A

From the figure, it is clear that,

$$\begin{gathered} \tan \theta =\frac{h}{a} \\ h=a\tan \theta \dots \dots \dots \dots \dots \dots \dots \left(1\right) \end{gathered}$$

Step 3: Determine the height from point B

$$\begin{gathered} \tan \left(90-\theta \right)=\frac{h}{b} \\ h=b\tan \left(90-\theta \right)\dots \dots \dots \dots \dots \dots \left(2\right) \end{gathered}$$

Step 4: Compute the height of the tower

Multiply, the equation $\left(1\right)$ and $\left(2\right)$,

$$\begin{gathered} h^2=ab\tan \theta ·cot\theta \\ h=\sqrt{ab} \end{gathered}$$

Hence, option B is the correct answer.