The angles of elevation of a cliff at a point A on the ground and a point B, 100 m vertically above A are `alpha and beta` respectively. The height of the cliff is

Options

A. `(100cotalpha)/(cotalpha-cotbeta)`
B. `(100cotbeta)/(cotalpha-cotbeta)`
C. `(100cotbeta)/(cotbeta-cotalpha)`
D. `(100cotbeta)/(cotbeta+cotalpha)`

Answer

To find the height of the cliff given the angles of elevation from two points, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Setup**:
- Let point A be on the ground and point B be 100 m vertically above A.
- Let the height of the cliff be DE, where D is the top of the cliff and E is the base of the cliff.
- The angles of elevation from points A and B to the top of the cliff D are α and β respectively.

2. **Identifying the Triangles**:
- In triangle ADE (where A is on the ground), we have:
\[ \tan(\alpha) = \frac{DE}{AD} \]
- In triangle BCE (where B is 100 m above A), we have:
\[ \tan(\beta) = \frac{DE - 100}{BC} \]

3. **Expressing the Height DE**:
- From triangle ADE:
\[ DE = AD \cdot \tan(\alpha) \]
- From triangle BCE:
\[ DE - 100 = BC \cdot \tan(\beta) \]
- Rearranging gives:
\[ DE = BC \cdot \tan(\beta) + 100 \]

4. **Relating AD and BC**:
- Since point B is directly above point A, we have:
\[ AD = BC \]
- Therefore, we can substitute BC with AD in the second equation:
\[ DE = AD \cdot \tan(\beta) + 100 \]

5. **Setting the Equations Equal**:
- Now we have two expressions for DE:
\[ AD \cdot \tan(\alpha) = AD \cdot \tan(\beta) + 100 \]

6. **Isolating AD**:
- Rearranging gives:
\[ AD \cdot \tan(\alpha) - AD \cdot \tan(\beta) = 100 \]
- Factoring out AD:
\[ AD (\tan(\alpha) - \tan(\beta)) = 100 \]
- Thus:
\[ AD = \frac{100}{\tan(\alpha) - \tan(\beta)} \]

7. **Finding the Height DE**:
- Substitute AD back into the equation for DE:
\[ DE = \frac{100}{\tan(\alpha) - \tan(\beta)} \cdot \tan(\alpha) \]
- Simplifying gives:
\[ DE = \frac{100 \tan(\alpha)}{\tan(\alpha) - \tan(\beta)} \]

8. **Final Height of the Cliff**:
- The height of the cliff is:
\[ \text{Height of the cliff} = DE = 100 \cdot \frac{\tan(\alpha)}{\tan(\alpha) - \tan(\beta)} \]