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 based on the angles of depression to two stations on opposite sides, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
- Let the height of the lighthouse be \( h \).
- Let the distance between the two stations be \( a \).
- The angles of depression from the top of the lighthouse to the two stations are \( \alpha \) and \( \beta \).

2. **Setting Up the Diagram**:
- Draw a vertical line representing the lighthouse of height \( h \).
- Mark the two stations on the ground level, one on the left and one on the right, such that the distance between them is \( a \).
- Let the distance from the base of the lighthouse to the left station be \( x \) and to the right station be \( y \). Thus, we have:
\[ x + y = a \]

3. **Using Trigonometry**:
- For the left station (angle of depression \( \alpha \)):
\[ \tan(\alpha) = \frac{h}{x} \implies x = \frac{h}{\tan(\alpha)} \]
- For the right station (angle of depression \( \beta \)):
\[ \tan(\beta) = \frac{h}{y} \implies y = \frac{h}{\tan(\beta)} \]

4. **Substituting the Values**:
- Substitute the expressions for \( x \) and \( y \) into the equation \( x + y = a \):
\[ \frac{h}{\tan(\alpha)} + \frac{h}{\tan(\beta)} = a \]

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

6. **Solving for \( h \)**:
- Rearranging gives:
\[ h = \frac{a}{\frac{1}{\tan(\alpha)} + \frac{1}{\tan(\beta)}} \]
- This can be simplified to:
\[ h = \frac{a \cdot \tan(\alpha) \cdot \tan(\beta)}{\tan(\alpha) + \tan(\beta)} \]

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