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 need to analyze the situation involving the ladder, the wall, and the angles given.

### Step 1: Understand the Setup
We have a ladder resting against a vertical wall at an angle \(\alpha\) with the horizontal. When the foot of the ladder is pulled away from the wall by a distance \(a\), the ladder slides down the wall by a distance \(b\) and makes a new angle \(\beta\) with the horizontal.

### Step 2: Identify the Length of the Ladder
Let \(L\) be the length of the ladder. The height of the ladder against the wall when at angle \(\alpha\) can be expressed as:
\[ h_1 = L \sin(\alpha) \]
The horizontal distance from the wall to the foot of the ladder is:
\[ d_1 = L \cos(\alpha) \]

### Step 3: New Position After Movement
After pulling the foot of the ladder away from the wall by distance \(a\), the new horizontal distance from the wall becomes:
\[ d_2 = d_1 + a = L \cos(\alpha) + a \]
The height of the ladder against the wall after sliding down by distance \(b\) becomes:
\[ h_2 = h_1 - b = L \sin(\alpha) - b \]

### Step 4: New Angle with the Horizontal
At the new position, the ladder makes an angle \(\beta\) with the horizontal. Therefore, we can express the new height and horizontal distance in terms of the angle \(\beta\):
\[ h_2 = L \sin(\beta) \]
\[ d_2 = L \cos(\beta) \]

### Step 5: Set Up the Equation
From the previous steps, we have two expressions for the height of the ladder:
\[ L \sin(\alpha) - b = L \sin(\beta) \]
And for the horizontal distance:
\[ L \cos(\alpha) + a = L \cos(\beta) \]

### Step 6: Solve for \(a\)
We can rearrange the equation for the horizontal distance to find \(a\):
\[ a = L \cos(\beta) - L \cos(\alpha) \]
Factoring out \(L\):
\[ a = L (\cos(\beta) - \cos(\alpha)) \]

### Step 7: Final Expression
Now, we have expressed \(a\) in terms of the length of the ladder and the angles:
\[ a = L (\cos(\beta) - \cos(\alpha)) \]