PQ is a post of height a, AB is a tower of height h at a distance x from the post, and `alpha and beta` are the angles of evevation of B, at P and Q respectively such that `alpha gt beta.` Then

Options

A. `h = x tan alpha `
B. `h = x sin 2 alpha `
C. `h = x cos 2 alpha`
D. `h = x sin alpha`

Answer

To solve the problem, we need to find the height of the tower (h) using the given information about the angles of elevation (α and β) and the height of the post (a).

### Step-by-Step Solution:

1. **Understand the Geometry**:
- We have a post PQ of height 'a'.
- There is a tower AB of height 'h' at a distance 'x' from the post PQ.
- The angles of elevation from points P and Q to point B (top of the tower) are α and β respectively, where α > β.

2. **Identify the Right Triangles**:
- From point P, we can form a right triangle PAB where:
- The height of the tower (AB) is 'h'.
- The distance from point P to the base of the tower (AP) is 'x'.
- From point Q, we can form another right triangle QAB where:
- The height of the tower (AB) is 'h'.
- The distance from point Q to the base of the tower (AQ) is 'x' (since Q is directly below P).

3. **Apply the Tangent Function**:
- For triangle PAB:
\[ \tan(\alpha) = \frac{h}{x} \]
- Rearranging gives:
\[ h = x \tan(\alpha) \]

4. **For Triangle QAB**:
- The height from point Q to point B can also be expressed as:
\[ \tan(\beta) = \frac{h - a}{x} \]
- Rearranging gives:
\[ h - a = x \tan(\beta) \]
- Thus:
\[ h = a + x \tan(\beta) \]

5. **Equating the Two Expressions for h**:
- We now have two expressions for h:
\[ h = x \tan(\alpha) \quad \text{(from triangle PAB)} \]
\[ h = a + x \tan(\beta) \quad \text{(from triangle QAB)} \]
- Setting them equal gives:
\[ x \tan(\alpha) = a + x \tan(\beta) \]

6. **Solve for h**:
- Rearranging gives:
\[ x \tan(\alpha) - x \tan(\beta) = a \]
- Factor out x:
\[ x (\tan(\alpha) - \tan(\beta)) = a \]
- Therefore:
\[ x = \frac{a}{\tan(\alpha) - \tan(\beta)} \]

7. **Substituting back to find h**:
- Substitute x back into the equation for h:
\[ h = x \tan(\alpha) = \frac{a \tan(\alpha)}{\tan(\alpha) - \tan(\beta)} \]

### Final Answer:
Thus, the height of the tower (h) is given by:
\[ h = \frac{a \tan(\alpha)}{\tan(\alpha) - \tan(\beta)} \]