From the top of a hill h metres high the angles of depression of the top and the bottom of a pillar are `alpha` and `beta` respectively. The height (in metres ) of the pillar is

Options

A. `(h(tan beta-tan alpha ))/(tan beta)`
B. `(h(tan alpha-tan beta))/(tan alpha )`
C. `(h (tan beta+tan alpha ))/(tan beta)`
D. `(h(tan beta+tan alpha ))/(tan alpha ) `

Explanation

To find the height of the pillar given the height of the hill and the angles of depression, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Geometry**:
- Let the height of the hill be \( h \) meters.
- Let the height of the pillar be \( x \) meters.
- The angles of depression from the top of the hill to the top and bottom of the pillar are \( \alpha \) and \( \beta \) respectively.

2. **Identify the Points**:
- Let point \( A \) be the top of the hill.
- Let point \( B \) be the bottom of the hill (ground level).
- Let point \( C \) be the top of the pillar.
- Let point \( D \) be the bottom of the pillar.

3. **Draw the Angles of Depression**:
- The angle of depression to the top of the pillar (point \( C \)) is \( \alpha \).
- The angle of depression to the bottom of the pillar (point \( D \)) is \( \beta \).

4. **Set Up the Right Triangles**:
- From point \( A \) to point \( C \):
- The height difference is \( h - x \).
- Let the horizontal distance from point \( A \) to the base of the pillar (point \( D \)) be \( y \).
- Using the tangent function, we have:
\[ \tan(\alpha) = \frac{h - x}{y} \]
- Rearranging gives:
\[ y = \frac{h - x}{\tan(\alpha)} \]

- From point \( A \) to point \( D \):
- The height difference is \( h \).
- The horizontal distance is still \( y \).
- Using the tangent function, we have:
\[ \tan(\beta) = \frac{h}{y} \]
- Rearranging gives:
\[ y = \frac{h}{\tan(\beta)} \]

5. **Equate the Two Expressions for \( y \)**:
- Since both expressions represent the same horizontal distance \( y \):
\[ \frac{h - x}{\tan(\alpha)} = \frac{h}{\tan(\beta)} \]

6. **Cross-Multiply and Solve for \( x \)**:
- Cross-multiplying gives:
\[ (h - x) \tan(\beta) = h \tan(\alpha) \]
- Expanding and rearranging:
\[ h \tan(\beta) - x \tan(\beta) = h \tan(\alpha) \]
\[ x \tan(\beta) = h \tan(\beta) - h \tan(\alpha) \]
- Finally, solving for \( x \):
\[ x = \frac{h (\tan(\beta) - \tan(\alpha))}{\tan(\beta)} \]

### Final Result:
The height of the pillar \( x \) is given by:
\[ x = h \left( \frac{\tan(\beta) - \tan(\alpha)}{\tan(\beta)} \right) \]