The angle of elevation of the top of a tower from two points at distance a and b metres from the base and in the same straight line with it are complementary, then height of the tower is :

Question

Accepted Answer

To determine the height of the tower when the angles of elevation from two points at distances $a$ and $b$ meters from the base are complementary, we need to use trigonometric principles.

### Explanation for Option 1: $\sqrt{ab}$

1. **Understanding Complementary Angles**: 
   - Complementary angles are two angles whose sum is $90^\circ$. If the angles of elevation from the two points are complementary, let's denote them as $\theta$ and $90^\circ - \theta$.

2. **Using Trigonometric Ratios**:
   - From the point at distance $a$, the angle of elevation is $\theta$. Therefore, the height $h$ of the tower can be expressed using the tangent function:
     $$
     \tan(\theta) = \frac{h}{a} \quad \Rightarrow \quad h = a \tan(\theta)
     $$
   - From the point at distance $b$, the angle of elevation is $90^\circ - \theta$. Using the tangent function again:
     $$
     \tan(90^\circ - \theta) = \frac{h}{b} \quad \Rightarrow \quad h = b \cot(\theta)
     $$

3. **Relating the Two Expressions**:
   - Since $\tan(90^\circ - \theta) = \cot(\theta)$, we have:
     $$
     \tan(\theta) \cdot \cot(\theta) = 1
     $$
   - Therefore, multiplying the two expressions for $h$:
     $$
     a \tan(\theta) = b \cot(\theta) \quad \Rightarrow \quad a \tan(\theta) = \frac{b}{\tan(\theta)}
     $$
   - Solving for $\tan^2(\theta)$:
     $$
     a \tan^2(\theta) = b \quad \Rightarrow \quad \tan^2(\theta) = \frac{b}{a}
     $$

4. **Finding the Height**:
   - Substitute $\tan(\theta) = \sqrt{\frac{b}{a}}$ into $h = a \tan(\theta)$:
     $$
     h = a \cdot \sqrt{\frac{b}{a}} = \sqrt{ab}
     $$

Thus, the height of the tower is $\sqrt{ab}$, which confirms that Option 1 is correct.

### Explanation for Why Other Options are Incorrect:

- **Option 2: $2\sqrt{ab}$**:
  - This option suggests that the height is twice $\sqrt{ab}$. However, the derivation shows that the height is exactly $\sqrt{ab}$, not twice that value. The factor of 2 is not supported by the trigonometric relationships derived.

- **Option 3: $ab$**:
  - This option implies a direct multiplication of the distances $a$ and $b$ without considering the trigonometric nature of the problem. The height depends on the tangent of the angles, not a simple product of distances.

- **Option 4: $\frac{1}{4}ab$**:
  - This option suggests a fraction of the product of $a$ and $b$. There is no basis in the trigonometric derivation for this fractional relationship. The height is determined by the geometric mean, not a fraction of the arithmetic product.

In conclusion, the correct height of the tower, given the conditions, is $\sqrt{ab}$, making Option 1 the correct choice.