The angles of elevation of the top of a tower h metre tall from two different points ont he same horizontal line are x and y `(x gt y)` . What is the distance between the points

Options

A. h (tan x - tan y)
B. `(h)/( tan x tan y)`
C. `(h (tan x - tan y))/(tan x tan y)`
D. `(h (tan x tan y))/( tan x - tan y)`

Answer

To find the distance between the two points from which the angles of elevation to the top of a tower are measured, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**:
We have a tower of height \( h \) meters. From two points \( A \) and \( B \) on the same horizontal line, the angles of elevation to the top of the tower are \( x \) and \( y \) respectively, where \( x > y \).

2. **Draw the Diagram**:
Draw a vertical line representing the tower, label the top of the tower as point \( C \) and the bottom as point \( D \). Mark points \( A \) and \( B \) on the horizontal line. The angle of elevation from point \( A \) to point \( C \) is \( x \) and from point \( B \) to point \( C \) is \( y \).

3. **Set Up the Right Triangles**:
In triangle \( ABC \) (where \( A \) is the point on the ground, \( B \) is directly below the tower, and \( C \) is the top of the tower):
- From point \( A \):
\[ \tan(x) = \frac{h}{d_A} \quad \text{(where \( d_A \) is the distance from \( A \) to the base of the tower)} \]
Thus,
\[ d_A = \frac{h}{\tan(x)} \]

- In triangle \( BCD \):
- From point \( B \):
\[ \tan(y) = \frac{h}{d_B} \quad \text{(where \( d_B \) is the distance from \( B \) to the base of the tower)} \]
Thus,
\[ d_B = \frac{h}{\tan(y)} \]

4. **Express the Distance Between Points**:
The distance \( d \) between points \( A \) and \( B \) can be expressed as:
\[ d = d_A - d_B \]

5. **Substitute the Values**:
Substitute the expressions for \( d_A \) and \( d_B \):
\[ d = \frac{h}{\tan(x)} - \frac{h}{\tan(y)} \]

6. **Combine the Fractions**:
To combine the fractions, find a common denominator:
\[ d = h \left( \frac{1}{\tan(x)} - \frac{1}{\tan(y)} \right) = h \left( \frac{\tan(y) - \tan(x)}{\tan(x) \tan(y)} \right) \]
Since \( x > y \), \( \tan(x) > \tan(y) \), thus:
\[ d = h \cdot \frac{\tan(x) - \tan(y)}{\tan(x) \tan(y)} \]

7. **Final Result**:
Therefore, the distance \( d \) between the two points is given by:
\[ d = h \cdot \frac{\tan(x) - \tan(y)}{\tan(x) \tan(y)} \]