From the top of a light house, the angles of depression of two stations on opposite sides of it at a distance a apart are `alpha` and `beta`. Find the height of the light house.

Answer

To find the height of the lighthouse given the angles of depression from its top to two stations on opposite sides, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Variables**:
- Let the height of the lighthouse be \( h \).
- Let the distance between the two stations be \( a \).
- Let the angle of depression to the first station be \( \alpha \).
- Let the angle of depression to the second station be \( \beta \).

2. **Draw the Diagram**:
- Draw a vertical line representing the lighthouse.
- Mark the top of the lighthouse as point \( O \) and the foot as point \( C \).
- Mark the two stations as points \( A \) and \( B \) such that the distance \( AB = a \).
- Draw horizontal lines from \( O \) to points \( A \) and \( B \) to represent the angles of depression \( \alpha \) and \( \beta \).

3. **Use Trigonometric Ratios**:
- In triangle \( AOC \):
\[ \tan(\alpha) = \frac{h}{AC} \]
Thus, we can express \( AC \) as:
\[ AC = \frac{h}{\tan(\alpha)} \]

- In triangle \( BOC \):
\[ \tan(\beta) = \frac{h}{BC} \]
Thus, we can express \( BC \) as:
\[ BC = \frac{h}{\tan(\beta)} \]

4. **Relate the Distances**:
- Since \( AC + BC = a \), we can substitute the expressions for \( AC \) and \( BC \):
\[ \frac{h}{\tan(\alpha)} + \frac{h}{\tan(\beta)} = a \]

5. **Factor Out \( h \)**:
- Taking \( h \) common from the left side:
\[ h \left( \frac{1}{\tan(\alpha)} + \frac{1}{\tan(\beta)} \right) = a \]

6. **Simplify the Equation**:
- The left side can be simplified using the formula for the sum of fractions:
\[ \frac{1}{\tan(\alpha)} + \frac{1}{\tan(\beta)} = \frac{\tan(\beta) + \tan(\alpha)}{\tan(\alpha) \tan(\beta)} \]
- Therefore, we have:
\[ h \cdot \frac{\tan(\beta) + \tan(\alpha)}{\tan(\alpha) \tan(\beta)} = a \]

7. **Solve for \( h \)**:
- Rearranging gives:
\[ h = \frac{a \cdot \tan(\alpha) \tan(\beta)}{\tan(\alpha) + \tan(\beta)} \]

### Final Result:
The height of the lighthouse \( h \) is given by:
\[ h = \frac{a \cdot \tan(\alpha) \tan(\beta)}{\tan(\alpha) + \tan(\beta)} \]