A ladder rests against a vertical wall at angle `alpha` to the horizontal . If is foot is pulled away from the wall through a distance 'a' so that it slides a distance 'b' down the wall making the angle `beta` with the horizontal , then a =

Options

A. `b tan ((alpha-beta)/2)`
B. `b tan ((alpha+beta)/2)`
C. `b cot ((alpha - beta)/2)`
D. none of these

Answer

To solve the problem step by step, we will analyze the situation involving the ladder, the wall, and the angles involved.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have a ladder resting against a vertical wall at an angle \( \alpha \) to the horizontal.
- The foot of the ladder is pulled away from the wall by a distance \( a \), causing the ladder to slide down the wall by a distance \( b \) and making a new angle \( \beta \) with the horizontal.

2. **Identifying Triangles**:
- Initially, when the ladder is at angle \( \alpha \), we can denote the length of the ladder as \( L \).
- The height of the ladder on the wall (vertical distance) can be expressed as \( h_1 = L \sin \alpha \).
- The horizontal distance from the wall to the foot of the ladder is \( d_1 = L \cos \alpha \).

3. **After the Ladder Slides**:
- After pulling the foot away and sliding down, the new height of the ladder on the wall becomes \( h_2 = L \sin \beta \).
- The new horizontal distance from the wall to the foot of the ladder is \( d_2 = L \cos \beta \).

4. **Relating the Distances**:
- The foot of the ladder is pulled away by a distance \( a \), which means:
\[ d_2 = d_1 + a \]
- The ladder slides down the wall by a distance \( b \), which gives us:
\[ h_1 - b = h_2 \]
- Therefore, we can express this as:
\[ L \sin \alpha - b = L \sin \beta \]

5. **Setting Up the Equations**:
- From the horizontal distance, we have:
\[ L \cos \beta = L \cos \alpha + a \]
- From the vertical distance, we can rearrange the equation:
\[ b = L \sin \alpha - L \sin \beta \]

6. **Finding the Value of \( a \)**:
- From the horizontal distance equation:
\[ a = L \cos \beta - L \cos \alpha \]
- This can be simplified to:
\[ a = L (\cos \beta - \cos \alpha) \]

### Final Answer:
Thus, the distance \( a \) that the foot of the ladder is pulled away from the wall is given by:
\[ a = L (\cos \beta - \cos \alpha) \]