The angle of elevation from two points at a distance of x and y from the feet of a Tower are complementary, the height of the Tower is

Options

A. `sqrt(xy)`
B. `x/y`
C. `sqrt(x/y)`
D. `sqrt(x+y)`

Answer

To find the height of the tower given that the angles of elevation from two points at distances \( x \) and \( y \) from the base of the tower are complementary, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Geometry**:
- Let the height of the tower be \( h \).
- Let the angle of elevation from the point at distance \( x \) be \( \theta \).
- Then, the angle of elevation from the point at distance \( y \) will be \( 90^\circ - \theta \) (since they are complementary).

2. **Set Up the Right Triangles**:
- From point A (at distance \( x \)):
\[ \tan(\theta) = \frac{h}{x} \quad \text{(1)} \]
- From point B (at distance \( y \)):
\[ \tan(90^\circ - \theta) = \cot(\theta) = \frac{h}{y} \quad \text{(2)} \]

3. **Express \(\tan(\theta)\) and \(\cot(\theta)\)**:
- From equation (1):
\[ h = x \tan(\theta) \quad \text{(3)} \]
- From equation (2):
\[ h = y \cot(\theta) \quad \text{(4)} \]

4. **Equate the Two Expressions for \( h \)**:
- From equations (3) and (4):
\[ x \tan(\theta) = y \cot(\theta) \]

5. **Use the Identity \(\cot(\theta) = \frac{1}{\tan(\theta)}\)**:
- Substitute \(\cot(\theta)\) in the equation:
\[ x \tan(\theta) = \frac{y}{\tan(\theta)} \]
- Multiply both sides by \(\tan(\theta)\):
\[ x \tan^2(\theta) = y \]

6. **Rearranging the Equation**:
- Rearranging gives:
\[ \tan^2(\theta) = \frac{y}{x} \]

7. **Substituting Back to Find \( h \)**:
- Substitute \(\tan^2(\theta)\) back into either equation (3) or (4). Using equation (3):
\[ h = x \tan(\theta) = x \sqrt{\frac{y}{x}} = \sqrt{xy} \]

8. **Final Result**:
- The height of the tower \( h \) is:
\[ h = \sqrt{xy} \]

### Conclusion:
Thus, the height of the tower is \( \sqrt{xy} \).