The angle of elevation of the top of a vertical tower from a point P on the horizontal ground was observed to be `alpha`. After moving a distance 2 meters from P towards the foot of the tower, the angle of elevation changes to `beta`. Then the height (in meters) of the tower is :

Options

A. `(2 sin alpha sin beta)/(sin (beta-alpha))`
B. `(sin alpha sin beta)/(cos (beta-alpha))`
C. `(2 sin (beta - alpha))/(sin alpha sin beta)`
D. `(cos (beta - alpha))/(sin alpha sin beta)`

Answer

To find the height of the tower given the angles of elevation and the distance moved, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem:**
- Let the height of the tower be \( h \).
- Let the distance from point P to the foot of the tower be \( l \).
- The angle of elevation from point P to the top of the tower is \( \alpha \).
- After moving 2 meters towards the tower, the angle of elevation becomes \( \beta \).

2. **Set Up the First Triangle (from Point P):**
- From point P, using the tangent function:
\[ \tan(\alpha) = \frac{h}{l + 2} \]
- Rearranging gives us:
\[ h = (l + 2) \tan(\alpha) \quad \text{(Equation 1)} \]

3. **Set Up the Second Triangle (from Point Q):**
- From point Q (which is 2 meters closer to the tower), the tangent function gives:
\[ \tan(\beta) = \frac{h}{l} \]
- Rearranging gives us:
\[ h = l \tan(\beta) \quad \text{(Equation 2)} \]

4. **Equate the Two Expressions for \( h \):**
- From Equation 1 and Equation 2, we have:
\[ (l + 2) \tan(\alpha) = l \tan(\beta) \]

5. **Rearranging the Equation:**
- Expanding and rearranging gives:
\[ l \tan(\beta) - l \tan(\alpha) = 2 \tan(\alpha) \]
- Factoring out \( l \):
\[ l (\tan(\beta) - \tan(\alpha)) = 2 \tan(\alpha) \]
- Therefore,
\[ l = \frac{2 \tan(\alpha)}{\tan(\beta) - \tan(\alpha)} \quad \text{(Equation 3)} \]

6. **Substituting \( l \) Back to Find \( h \):**
- Substitute Equation 3 into Equation 2:
\[ h = \left(\frac{2 \tan(\alpha)}{\tan(\beta) - \tan(\alpha)}\right) \tan(\beta) \]
- Simplifying gives:
\[ h = \frac{2 \tan(\alpha) \tan(\beta)}{\tan(\beta) - \tan(\alpha)} \]

7. **Final Expression for Height \( h \):**
- Thus, the height of the tower is:
\[ h = \frac{2 \tan(\alpha) \tan(\beta)}{\tan(\beta) - \tan(\alpha)} \]