The angle of elevation of top of a house from top and bottom of tree are respectively x and y. If height of the tree is h meter then what is the height of the house ?

Options

A. `(hcotx)/(cotx+coty)`
B. `(hcoty)/(cotx+coty)`
C. `(hcotx)/(cotx-coty)`
D. `(hcoty)/(cotx-coty)`

Answer

To find the height of the house given the angles of elevation from the top and bottom of a tree, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
- Let the height of the tree be \( h \) meters.
- Let the angle of elevation from the top of the tree to the top of the house be \( x \).
- Let the angle of elevation from the bottom of the tree to the top of the house be \( y \).
- We need to find the height of the house, which we will denote as \( AB \).

2. **Setting Up the Diagram**:
- Draw a vertical line representing the tree (height \( h \)).
- Draw another vertical line representing the house (height \( AB \)).
- Mark the points:
- \( A \) as the top of the house,
- \( B \) as the bottom of the house,
- \( C \) as the bottom of the tree,
- \( D \) as the top of the tree.
- The angles \( \angle DAB = x \) and \( \angle CAB = y \).

3. **Using Trigonometric Ratios**:
- From triangle \( ADE \) (where \( E \) is a point on the ground directly below \( A \)):
- The cotangent of angle \( x \) can be expressed as:
\[ \cot x = \frac{DE}{AE} \]
Here, \( DE \) is the horizontal distance from the tree to the house, and \( AE \) is the height from the bottom of the house to the top of the tree, which we denote as \( p \).

- From triangle \( ABC \):
- The cotangent of angle \( y \) can be expressed as:
\[ \cot y = \frac{BC}{AB} \]
Here, \( BC \) is the same horizontal distance as \( DE \) and \( AB \) is the height of the house.

4. **Setting Up the Equations**:
- From triangle \( ADE \):
\[ DE = p \cdot \cot x \]
- From triangle \( ABC \):
\[ DE = h + p \cdot \cot y \]

5. **Equating the Two Expressions for DE**:
- Since both expressions represent the same horizontal distance \( DE \):
\[ p \cdot \cot x = h + p \cdot \cot y \]

6. **Rearranging the Equation**:
- Rearranging gives:
\[ p \cdot \cot x - p \cdot \cot y = h \]
- Factoring out \( p \):
\[ p(\cot x - \cot y) = h \]
- Solving for \( p \):
\[ p = \frac{h}{\cot x - \cot y} \]

7. **Finding the Height of the House**:
- The height of the house \( AB \) is given by:
\[ AB = h + p \]
- Substituting \( p \):
\[ AB = h + \frac{h}{\cot x - \cot y} \]
- Finding a common denominator:
\[ AB = \frac{h(\cot x - \cot y) + h}{\cot x - \cot y} = \frac{h \cot x}{\cot x - \cot y} \]

### Final Answer:
The height of the house is given by:
\[ AB = \frac{h \cot x}{\cot x - \cot y} \]