The angles of elevation of the top of a tower from two points at distances m and n meters are complementary. If the two points and the base of the tower ae on the same straight line, then the height of the tower is
(a)`sqrt(mn)`
(b)mn
(c)`m/n`
(d)None of these

Options

A. `sqrt(mn)`
B. mn
C. `m/n`
D. None of these

Answer

To solve the problem, we need to find the height of the tower given that the angles of elevation from two points at distances \( m \) and \( n \) meters from the base of the tower are complementary.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- Let \( h \) be the height of the tower.
- Let \( \alpha \) be the angle of elevation from the point at distance \( m \) meters.
- Let \( \beta \) be the angle of elevation from the point at distance \( n \) meters.
- According to the problem, \( \alpha + \beta = 90^\circ \).

2. **Using Trigonometric Ratios**:
- From the point at distance \( m \):
\[ \tan(\alpha) = \frac{h}{m} \]
- From the point at distance \( n \):
\[ \tan(\beta) = \frac{h}{n} \]

3. **Using the Complementary Angle Identity**:
- Since \( \alpha + \beta = 90^\circ \), we have:
\[ \beta = 90^\circ - \alpha \]
- Therefore, using the identity \( \tan(90^\circ - \theta) = \cot(\theta) \):
\[ \tan(\beta) = \cot(\alpha) = \frac{1}{\tan(\alpha)} \]

4. **Setting Up the Equation**:
- From the above, we can write:
\[ \tan(\beta) = \frac{n}{h} \quad \text{and} \quad \tan(\alpha) = \frac{h}{m} \]
- Thus, we have:
\[ \frac{h}{n} = \frac{1}{\frac{h}{m}} \implies \frac{h}{n} = \frac{m}{h} \]

5. **Cross Multiplying**:
- Cross multiplying gives:
\[ h^2 = mn \]

6. **Finding the Height**:
- Taking the square root of both sides:
\[ h = \sqrt{mn} \]

### Conclusion:
The height of the tower is \( \sqrt{mn} \).

### Answer:
(a) \( \sqrt{mn} \)