A tree standing on a horizontal plane is leaning towards East. At two points situated at distances a and b exactly due West of it, the angle of elevation of the top are respectively `alpha and beta`. Height of the top from the ground is
`((b-a)tan alpha tan beta)/(tan alpha+ tan beta)`

Answer

To solve the problem, we need to determine the height of the tree leaning towards the east based on the angles of elevation observed from two points located due west of the tree. We will denote the height of the tree as \( H \), the distance from the tree to the first observation point as \( A \), and the distance to the second observation point as \( B \). The angles of elevation from these points are \( \alpha \) and \( \beta \), respectively.

### Step-by-Step Solution:

1. **Understanding the Geometry**:
- Let \( T \) be the top of the tree, \( O \) be the base of the tree, and \( P \) and \( Q \) be the points where the angles of elevation are measured.
- The distance from the tree to point \( P \) is \( A \) and to point \( Q \) is \( B \).
- The angles of elevation from points \( P \) and \( Q \) are \( \alpha \) and \( \beta \).

2. **Setting Up the Triangles**:
- From point \( P \):
\[ \tan(\alpha) = \frac{H}{A} \]
Rearranging gives:
\[ H = A \tan(\alpha) \quad \text{(1)} \]

- From point \( Q \):
\[ \tan(\beta) = \frac{H}{B} \]
Rearranging gives:
\[ H = B \tan(\beta) \quad \text{(2)} \]

3. **Equating the Two Expressions for \( H \)**:
- From equations (1) and (2):
\[ A \tan(\alpha) = B \tan(\beta) \]
- Rearranging gives:
\[ \frac{H}{\tan(\alpha)} = A \quad \text{and} \quad \frac{H}{\tan(\beta)} = B \]

4. **Finding the Height \( H \)**:
- We can express \( H \) in terms of \( A \) and \( B \):
\[ H = \frac{A \tan(\alpha) B \tan(\beta)}{\tan(\alpha) + \tan(\beta)} \]

5. **Final Expression for Height**:
- The height \( H \) of the tree is given by:
\[ H = \frac{(B - A) \tan(\alpha) \tan(\beta)}{\tan(\alpha) + \tan(\beta)} \]

### Final Result:
Thus, the height of the tree from the ground is:
\[ H = \frac{(B - A) \tan(\alpha) \tan(\beta)}{\tan(\alpha) + \tan(\beta)} \]